Fixed-Points of Social Choice: An Axiomatic Approach to Network Communities
Christian Borgs, Jennifer Chayes, Adrian Marple, Shang-Hua Teng
aa r X i v : . [ c s . S I] O c t Fixed-Points of Social Choice: An Axiomatic Approach toNetwork Communities
Christian BorgsMicrosoft Research Jennifer ChayesMicrosoft Research Adrian MarpleStanford Shang-Hua Teng ∗ USCOctober 11, 2018
Abstract
We provide the first social choice theory approach to the question of what constitutes acommunity in a social network. Inspired by social choice theory in voting and other contexts[2], we start from an abstract social network framework, called preference networks [3]; theseconsist of a finite set of members and a vector giving a total ranking of the members in theset for each of them (representing the preferences of that member).Within this framework, we axiomatically study the formation and structures of commu-nities. Our study naturally involves two complementary approaches. In the first, we applysocial choice theory and define communities indirectly by postulating that they are fixedpoints of a preference aggregation function obeying certain desirable axioms. In the sec-ond, we directly postulate desirable axioms for communities without reference to preferenceaggregation, leading to a natural set of eight community axioms.These two approaches allow us to formulate and analyze community rules. We prove ataxonomy theorem that provides a structural characterization of the family of those commu-nity rules that satisfies all eight axioms. The structure is actually quite beautiful: the familysatisfying all eight axioms forms a bounded lattice under the natural intersection and unionoperations of community rules. The taxonomy theorem also gives an explicit characterizationof the most comprehensive community rule and the most selective community rule consis-tent with all community axioms. This structural theorem is complemented with a complexityresult : we show that while identifying a community by the selective rule is straightforward,deciding if a subset satisfies the comprehensive rule is coNP-complete. Our studies also shedlight on the limitations of defining community rules solely based on preference aggregation.In particular, we show that many aggregation functions lead to communities which violateat least one of our community axioms. These include any aggregation function satisfyingArrow’s independence of irrelevant alternative axiom as well as commonly used aggregationschemes like the Borda count or generalizations thereof. Finally, we give a polynomial-timerule consistent with seven axioms and weakly satisfying the eighth axiom. ∗ Supported in part by NSF grants CCF-1111270 and CCF-0964481 and by a Simons Investigator Award fromthe Simons Foundation. Introduction: Formulating Preferences and Communities
A fundamental problem in network analysis is the characterization and identification of subsetsof nodes in a network that have significant structural coherence. This problem is usually studiedin the context of community identification and network clustering.
Like other inverse problemsin machine learning, this one is conceptually challenging: There are many possible ways tomeasure the degree of coherence of a subset and many possible interpretations of affinities tomodel network data. As a result, various seemingly reasonable/desirable conditions to qualifya subset as a community have been studied in the literature [15, 21, 18, 11, 3, 10, 8, 17, 22, 6].The fact that there are an exponential number of candidate subsets to consider makes directcomparison of different community characterizations quite difficult.Among the challenges in the study of communities in a social and information network arethe following two basic mathematical problems: • Extension of individual affinities/preferences to community coherence : A (so-cial) network usually represents pairwise interactions among its members, while the notionof communities is defined over its larger subsets. Thus, to model the formation of com-munities, we need a set of consistent rules to extend the pairwise relations or individualpreferences to community coherence. • Inference of missing links : Since networks typically are sparse, we also need methodsto properly infer the missing links from the given network data.In this paper, we take what we believe is a novel and principled approach to the problem ofcommunity identification. Inspired by the classic work in social choice theory [2], we propose anaxiomatic approach towards understanding network communities, providing both a frameworkfor comparison of different community characterizations, and relating community identificationto well-studied problems in social choice theory [2]. Here, we focus on the problem of definingcommunity rules and coherence measures from individual preferences presented in the inputsocial/information network, but we think that this study will also provide the foundation for anaxiomatic approach to the problem of inferring missing links. We plan to address this secondproblem in a subsequent paper, which will use this paper as a foundation.Through the lens of axiomatization, we examine both mathematical and complexity-theoreticstructures of communities that satisfy a community rule or a set of community axioms. We alsostudy the stability of network communities, and design algorithms for identifying and enumer-ating communities with desirable properties.While the study initiated here is conceptual, we believe it will ultimately enable a moreprincipled way to choose among community formation models for interpretation of current ex-periments, and also suggest future experiments.
Before presenting the highlights of our work, we first define an abstract social network frameworkwhich enables us to focus on the axiomatic study of community rules. This framework is inspiredby social choice theory [2] and was first used in [3] in the context of community identification formodeling social networks with complete preference information. We will refer to each instanceof this framework as a preference network . Below, for a non-empty finite set V , let L ( V ) denote2he set of all linear orders on V , represented, e.g., by the set of all bijections π : V → [1 : | V | ],where as usual, [ n : m ] is the set { n, n + 1 , , . . . , m } . Alternatively, π can be represented by theordered list π = [ x , x , . . . , x | V | ], where x i ∈ V is such that π ( x i ) = i ; in our notation, π ( x )thus represents the rank of x in the ordered list π = [ x , x , . . . , x | V | ]. Definition 1 (Preference Networks) . A preference network is a pair A = ( V, Π) , where V is anon-empty finite set and Π is a preference profile on V , defined as an element Π = { π i } i ∈ V ∈ L ( V ) V . Here π i specifies the total ranking of V in the order of i ’s preference: ∀ s, u, v ∈ V , s prefers u to v , denoted by u ≻ π s v , if and only if π s ( u ) < π s ( v ) . As argued in [3], a real-life social network may be viewed as sparse, observed social interac-tions of an underlying latent preference network. In this view, the communities of a preferencenetwork may be considered to be the ground truth set of potential communities in its observedsocial network.
Our main contribution is an axiomatic framework for studying community formation in prefer-ence networks, and mathematical, complexity-theoretic, and algorithmic investigation of com-munity structures in this framework. Our work on axiomatization of network communities canbe organized into two related parts: (1) communities as fixed points of social choice aggregationfunctions; and (2) communities via direct axiomatic characterization. In the second part, wespecify eight axioms we would like the communities to obey, and find conditions under whichsuch communities exist. In the first part, we specify social choice aggregation functions forwhich the communities will be fixed points; this first method allows for an “indirect” axiomaticcharacterization in that the aggregation functions themselves could be taken to obey axioms[2, 24], which would then indirectly characterize the communities which result as fixed points.
Communities as fixed points of social choice
Our approach of starting from preference networks to study communities naturally connectscommunity formation to social choice theory [2], which provides a theoretical framework forunderstanding the problem of combining individual preferences into a collective preference ordecision. In this first part of our analysis, we use preference aggregation functions studied insocial choice theory [2] to characterize communities by defining communities as fixed points ofa preference aggregation function.Since real-world voting schemes and preference aggregation functions do not always producea total order, we will use the following notation in the definition below. Let L ( V ) denote theset of all ordered partitions of V . For a σ ∈ L ( V ), for i, j ∈ V , we use i ≻ σ j to denote that i is strictly preferred to j (that is, i and j belong to different partitions, and the partition containing In broader settings, one may want to consider preferences that allow indifference or partially ordered prefer-ences, or both. One may also model a social network by a cardinal affinity network that specify each member’spreference by a weighted affinity vector, for example with weights from [0 ,
1] where 1 and 0, respectively, representthe highest and lowest preferences. To distinguish ordinal and cardinal preferences, we refer to the latter as an affinity network . Both models are referred to as affinity systems in [3]. For simplicity of exposition, we first focuson preference networks. In Section 7, we discuss the possible extension of our framework. is ahead of the partition containing j in σ ). In this case, we also say j ≺ σ i . We will use i (cid:23) σ j to denote that i ≻ σ j or i and j are in the same partition.To continue, we need some notions motivated by social choice theory. In this context, V will be considered a set of “candidates”. We’ll also need a set of possible voters, S , whichis assumed to be a countable set – if not otherwise specified, we identify S with the positiveintegers N . With a slight abuse of notation, we denote the union of L ( V ) S over all non-emptyfinite S ⊆ S by L ( V ) ∗ . A preference aggregation function is then defined to be an arbitraryfunction F : L ( V ) ∗ → L ( V ). Given a non-empty finite set of voters S and a preference profileΠ S = { π s : s ∈ S } ∈ L ( V ) S , the image F (Π S ) is called the aggregated preference of S . Definition 2. (Communities as Fixed Points of Social Choice)
Let A = ( V, Π) be apreference network, F : L ( V ) ∗ → L ( V ) be a preference aggregation function, and ∅ 6 = S ⊆ V . S is called a community of A with respect to F if and only if u ≻ F (Π S ) v , ∀ u ∈ S, v ∈ V − S .The function C F mapping A into the set of communities defined above is called the fixedpoint rule with respect to F . If F is not specified, i.e., if there exists an F such that C = C F ,we call C simply a fixed point rule . Informally, this definition says that a community is a subset S ⊆ V such that, when we aggre-gate the preferences of all its members, the resulting aggregated preference puts the members S as the top | S | elements. In other words, under the aggregation function F , the members of thecommunity “vote” for themselves. Thus, S is a fixed point of its aggregated preference. The com-munity characterization of Definition 2 generalizes the following concept of self-determinationof [3]: Definition 3. (B CT Communities)
Let A = ( V, Π) be a preference network. For ∅ 6 = S ⊆ V and i ∈ V , let φ Π S ( i ) denote the number of votes that member i would receive if each member s ∈ S was casting one vote for each of its | S | most preferred members according to its preference π s .In other words, φ Π S ( i ) = |{ s : ( s ∈ S ) & ( π s ( i ) ∈ [1 : | S | ]) }| . Then, S is B CT-self-determined if everyone in S receives more votes from S then everyone outside S . It is easy to see that the B CT voting rule is an instance of a fixed-point rule, with preferenceaggregation function F defined by v ≻ F (Π S ) w iff φ Π S ( v ) > φ Π S ( w ).We will also refer to a community according to Definition 2 as an F -self-determined commu-nity . We are particularly interested in those aggregation functions that satisfy various axioms insocial choice theory [2], since this enables us to utilize established social choice theory to studyall conceivable self-determination community rules within one unified framework. For example,it allows us to reduce the fairness analysis for community formation to the fairness of preferenceaggregation functions.Arrow’s celebrated impossiblility theorem and subsequent work in social choice theory [2]point to both challenges and exciting opportunities for understanding communities in prefer-ence networks. Recall that Arrow’s theorem states that for n >
2, no (strictly linear) preferenceaggregation function satisfies all of the following three axiomatic conditions:
Unanimity , Inde-pendence of Irrelevant Alternatives , and
Non-Dictatorship (see Section 2 for definitions.) On the Note that in our notation without further requiring F to satisfy additional conditions, the labels in S matter:e.g., even if π and π are the same permutation of [ n ], the values of F (Π { } ) and F (Π { } ) can be different. In the case of ties, we allow for ties among the top | S | members, as well as among the lower ranked members,but not between the top | S | members and anyone below. Communities via direct axiomatic characterization
In this second approach, we will use a more direct axiomatic characterization to study networkcommunities. To this end, we use a set-theoretical community function as a means to characterizea community rule.
Definition 4 (Community Functions) . Let A denote the set of all preference networks. A community function is a function C that maps a preference network A = ( V, Π) to a characteristicfunction of non-empty subsets of V . In other words, C ( A ) ⊆ V −{∅} is an indicator function of V −{∅} . We say a subset S ⊆ V is a community in a preference network A = ( V, Π) accordingto a community function C if and only if S ∈ C ( A ) . To simplify our notation, for A = ( V, Π) we often write C ( V, Π) instead of C (( V, Π)) . We use axioms to state properties, such as fairness and consistency, that a desirable commu-nity function should have when applied to all preference networks. An example is the propertythat the community function should be isomorphism-invariant: Here an isomorphism betweentwo preference networks A = ( V, Π) and A ′ = ( V ′ , Π ′ ) is a bijection σ : V → V ′ such thatΠ ′ = σ (Π), i.e., such that for all s, v ∈ V , π ′ σ ( s ) ( σ ( v )) = π s ( v ), and two preference networks A and A ′ are isomorphic to each other if there exists such an isomorphism. Isomorphism invariancethen requires that for any pair of isomorphic preference networks A = ( V, Π) and A ′ = ( V ′ , Π ′ )and any isomorphism σ between A and A ′ , if S ⊂ V is a community in A , then σ ( S ) should stillbe a community in the A ′ . Another example is the property of monotonic characterization : If S is a community in A = ( V, Π), then S should remain a community in every preference network A ′ = ( V, Π ′ ) such that for all u, s ∈ S and v ∈ V , if u ≻ π s v then u ≻ π ′ s v .In Section 2, we propose a natural set of eight desirable community axioms. Six of them, in-cluding both examples above, provide a positive characterization of communities. These axiomsconcern the consistency, fairness, and robustness of a community function, as well as the com-munity structures when a preference network is embedded in a larger preference network. Theother two axioms address the necessary stability and self-approval conditions that a communityshould satisfy. Constructing and Analyzing Community Rules
While Definition 4 is convenient for the study of the mathematical structure of our theory,community identification is a computational problem as much as a mathematical problem. Thus,it is desirable that communities can be characterized by a constructive community function C that is: • Consistent : C satisfies all (or nearly all) axioms; • Constructive : Given a preference network A = ( V, Π), and a subset S ⊆ V , one candetermine in polynomial-time (in n = | V | ) if S ∈ C ( A ).5 Samplable : One can efficiently obtain a random sample of C ( A ). • Enumerable : One can efficiently enumerate C ( A ), for instance, in time O ( n k · |C ( A ) | ) fora constant k .Our two axiomatic approaches allow us to formulate a rich family of community rules and analyzetheir properties. Using the fixed-point rule, we can define a constructive community functionbased on any polynomial-time computable aggregation function. Alternatively, we can use oneaxiom or a set of axioms as a community rule. We can also define a community rule by theintersection of a fixed-point rule and a set of axioms. In this paper, we aim to characterize thecommunity rules that satisfy a set of “reasonable” axioms , and address the basic questions: • Is there an aggregation function leading to a community rule satisfying this setof “reasonable” axioms? • What is the complexity of the community rules based on these axioms? • How are different community rules satisfying our axioms related to each other?For example, given two community rules C and C satisfying our axioms, doesthe rule C defined by C ( A ) := C ( A ) ∩ C ( A ) obey our axioms as well? Structural and Complexity-Theoretic Results
Our main structural result is a taxonomy theorem that provides a complete characterizationof the most comprehensive community rule and the most selective community rule consistentwith all our community axioms. This result illustrates an interesting contrast to the classicaxiomatization result of Arrow [2] and the more recent result of Kleinberg on clustering [9]that inspired our work. Unlike voting or clustering where the basic axioms lead to impossiblitytheorems, the preference network framework offers a natural community rule, which we call the
Clique Rule, that is intuitively fair, consistent, and stable, although selective (See Section 4for more details): S is a community according the Clique Rule iff each member of S prefersevery member of S over every non-member. Indeed the Clique Rule satisfies all our axioms.Our analysis then leads us to a community rule which is consistent with all axioms – we callit the Comprehensive Rule – such that for any community rule C satisfying all axioms and allpreference network A , C clique ( A ) ⊆ C ( A ) ⊆ C comprehensive ( A ). Perhaps more interesting, underthe natural operations of union and intersections, the set of all community rules satisfying allour axioms becomes a lattice with C clique ( A ) and C comprehensive ( A ) forming a lower and upperbound, respectively.We complement this structural theorem with a complexity result: we show that while iden-tifying a community by the Clique Rule is straightforward, it is coNP-complete to determine ifa subset satisfies the comprehensive rule.Our studies also shed light on the limitations of formulating community rules solely basedon preference aggregation. In particular, we show that many aggregation functions lead tocommunities which violate at least one of our community axioms. We give two impossibility-liketheorems.1. Any fixed-point rule based on commonly used aggregation schemes like Borda count orgeneralizations thereof – such as the B CT self-determination rule – is inconsistent with(at least) one of our axioms. 6. For any aggregation function satisfying Arrow’s independence of irrelevant alternativeaxiom, its fixed-point rule must violate one of our axioms.Finally, using our direct axiomatic framework, we analyze the following natural constructivecommunity function inspired by preference aggregations.
Definition 5 (Harmonious Communities) . A non-empty subset S ⊆ V is a harmonious com-munity of a preference network A = ( V, Π) if for all u ∈ S and v ∈ V − S , the majority of { π s : s ∈ S } prefer u over v . We will show that the harmonious community rule is consistent with seven axioms andsatisfies a weaker form of the eighth axiom. In addition, various stable versions of harmoniouscommunities (see the discussion below) enjoy some degree of samplablility and enumerability.
Stability of Communities and Algorithms
In real-world social interactions, some communities are more stable or durable than otherswhen people’s interests and preferences evolve over time. For example, some music bands staytogether longer than others. Inspired by the work of [3] and Mishra et al. [11] on modeling thisphenomenon, we examine the impact of stability on the community structure.To motivate our discussion, we first recall the main definition and result of [3]:
Definition 6.
For ≤ β < α ≤ , a non-empty subset S ⊆ V is an ( α, β ) − B CT community in A = ( V, Π) iff φ Π S ( u ) ≥ α · | S | ∀ u ∈ S and φ Π S ( v ) < β · | S | ∀ v S . It was shown in [3] that, in any preference network, there are only polynomially many stableB CT communities when the parameters α, β are constants, and they can be enumerated inpolynomial time, showing that the strength of community coherence has both structural andcomputational implications.In Section 6, we consider several stability conditions in our axiomatic community framework.In one direction, we examine the structure of the communities (defined by a fixed-point commu-nity rule) that remain self-determined even after a certain degree of perturbation in its members’preferences. In this context, for example, we can reinterpret the B CT-stability defined above asfollows: A subset S ⊆ V is an ( α, β ) − B CT community in a preference network A if it remainsself-determined when | S | · ( α − β ) / S make arbitrary changes to their preferences.In the other direction, we consider some notions of stability derived directly from the social-choice based community framework where members of a community separate themselves fromthe rest. We can further use the separability as a measure of the community strength and sta-bility to capture the intuition that stronger communities are also themselves more integrated.As a concrete example, we show in Section 6 that there are a quasi-polynomial number of stableharmonious communities for all these notions of stability. This result demonstrates that thereexists a constructive community function that essentially satisfies all our axioms, whose stablecommunities are quasi-polynomial-time samplable and enumerable. In this section, we define our eight core axioms, give a more formal treatment of social choiceaxioms, and examine several properties of community rules and the relations these have witheach other. 7 .1 Lexicographic Preference
The following notion will be crucial in several parts of this paper, and is implicitly used in ourfirst two axioms below.
Definition 7 (Lexicographical Preferences) . Given a preference network ( V, Π) and two non-empty disjoint subsets G and G ′ of equal size, we say that s ∈ V lexicographically prefers G ′ over G if there exists a bijection f s : G → G ′ such that f s ( u ) ≻ π s u for all u ∈ G .We say that a group T ⊂ V lexicographically prefers G ′ over G if every member s ∈ T lexicographically prefers G ′ to G , i.e., if there exists a set of bijections { f s : G → G ′ | s ∈ T } such that f s ( u ) ≻ π s u for all u ∈ G and all s ∈ T . Note that, in contrast to the standard lexicographical order, lexicographical preference isonly a partial order. The notion is motivated by the following proposition.
Proposition 1.
Let π ∈ L ( V ) , let G and G ′ be disjoint subsets of V with | G | = | G ′ | . Let G [ i ] (and G ′ [ i ] ) be the i th highest ranked element of G (and G ′ ) according to π . Then there existsa bijection f : G −→ G ′ such that for all g ∈ G , f ( g ) ≻ π g if and only for all i ∈ [1 : | G | ] , G ′ [ i ] ≻ π G [ i ] .Proof. Suppose f satisfies the condition of the proposition. Then G ′ [1] (cid:23) π f ( G [1]) ≻ π G [1].If G ′ [1] = f ( G [1]), define h to be the bijection on G ′ which exchanges G ′ [1] and f ( G [1]), anddefine ˜ f = g ◦ f . Then G ′ [1] = ˜ f ( G [1]) ≻ π G [1] while ˜ f still satisfies the condition of theproposition. Removing G [1] from G and G ′ [1] from G ′ , we continue by induction to prove theonly if statement. The if statement is obvious - just define f by f ( G [ i ]) = G ′ [ i ]. For the following definitions, fix a ground set V and a community function C . Axiom 1 ( Group Stability (GS)) . If Π is a preference profile over V and S ∈ C ( V, Π) , then S is group stable with respect to Π . Here a subset S ⊂ V is called group stable with respect to Π if for all non-empty G ( S , all G ′ ⊂ V − S of the same size as G , and all tuples of bijections, ( f i : G → G ′ , i ∈ S − G ) , there exists s ∈ S − G , u ∈ G such that u ≻ π s f s ( u ) . This axiom provides a type of game-theoretic stability [14, 13, 4, 19, 20], and states thatno subgroup in a community can be replaced by an equal-size group of non-members that arelexicographically preferred by the remainder of the community members. For instance, if thesubgroup is of size 1, this means that there is no outsider that is universally preferred to thismember, excluding that member’s own opinion. On the other end of the spectrum, if thesubgroup is all but one person, then group stability states that there must be someone fromthat member’s top choices, and thus represents a type of individual rationality condition. Notethat the set V is vacuously group stable for all Π. Axiom 2 ( Self-Approval (SA)) . If Π is a preference profile over V , and S ∈ C ( V, Π) then S is self-approving with respect to Π . Here a subset S ⊂ V is called self-approving with respectto Π if for all G ′ ⊆ V − S of the same size as S , and all tuples of bijections ( f i : S → G ′ , i ∈ S ) there exists s, u ∈ S , such that u ≻ π s f s ( u ) . SA uses the same partial ordering of groups as the first, and requires that there is nooutside group of the same size as S which is lexicographically preferred to S by everyone in S .It generalizes the intuition that a singleton should be a community only if that member prefersherself to everyone else. Note that any set S of size larger than | V | / Axiom 3 ( Anonymity (A)) . Let
S, S ′ ⊂ V and Π , Π ′ be such S ′ = σ ( S ) and Π ′ = σ (Π) for somepermutation σ : V → V . Then S ∈ C ( V, Π) ⇐⇒ S ′ ∈ C ( V, Π ′ ) . A staple axiom,
Anonymity , states that labels should have no effect on a community function.
Axiom 4 ( Monotonicity (Mon)) . Let S ⊂ V . If Π and Π ′ are such that for all s ∈ Su ≻ π ′ s v = ⇒ u ≻ π s v for all u ∈ S, v ∈ V then S ∈ C ( V, Π ′ ) = ⇒ S ∈ C ( V, Π) . The Axiom
Monotonicity states that, if a member of a community gets promoted withoutnegatively impacting other members, then that subset must remain a community. Thus thisaxiom reflects the fact that high positions imply greater affinities towards those people. Notethat
Mon also allows non-members to change arbitrarily, as long as their positions relative toany members remains the same or worse.
Axiom 5 ( Coherence Robustness of Non-Members (CRNM)) . Let S ⊂ V . If Π and Π ′ are suchthat for all s, t ∈ S v ≻ π ′ s w ⇐⇒ v ≻ π ′ t w for all v, w / ∈ S and π ′ s ( u ) = π s ( u ) for all u ∈ S, then S ∈ C ( V, Π ′ ) = ⇒ S ∈ C ( V, Π) . Axiom 6 ( Coherence Robustness of Members (CRM)) . Let S ⊂ V . If Π and Π ′ are such that forall s, t ∈ S we have u ≻ π ′ s w ⇐⇒ u ≻ π ′ t w for all u, w ∈ S and π ′ s ( v ) = π s ( v ) for all v / ∈ S, then S ∈ C ( V, Π ′ ) = ⇒ S ∈ C ( V, Π) . The two
Coherence Robustness
Axioms reflect the fact that, if community members agreeabout their preferences concerning either members or non-members, they are less likely to bea community. In the case of non-members, agreement implies that some non-member is morepreferred and therefore more likely to break up the community. Contrariwise, in the case ofmembers, agreement implies some member is less preferred and more likely to be ousted.
Axiom 7 ( World Community (WC)) . For all preference profiles Π , V ∈ C ( V, Π) .
9o state the next axiom, we define the projection A | V ′ of a preference network A = ( V, Π)onto a subset V ′ ⊂ V as the preference network A | V ′ = ( V ′ , Π | V ′ ) where Π | V ′ = { π ′ s } s ∈ V ′ isdefined by setting π ′ s to be the linear order on L ( V ′ ) which keeps the relative ordering of allmembers of V ′ , i.e., for all s, u, v ∈ V ′ , u ≻ π ′ s v ⇐⇒ u ≻ π s v . We say that A ′ is embedded into A if A ′ = A | V ′ for some V ′ ⊂ V . Axiom 8 ( Embedding (Emb)) . If A ′ = ( V ′ , Π ′ ) is embedded into A = ( V, Π) and π i ( j ) = π ′ i ( j ) for all i, j ∈ V ′ then C ( A ′ ) = C ( A ) ∩ V ′ . In other words, if a network ( V ′ , Π ′ ) is embedded into a larger network ( V, Π) in such a waythat, with respect to the preferences in the larger network, the members of the smaller networkprefer each other over everyone else, then the set of communities in the larger network whichare subsets of V ′ is identical to the set of communities in the smaller network.Note that, in contrast to the first seven axioms, which refer to a fixed finite ground set V ,the last axiom links different grounds sets to each other. Strictly speaking, a community rule C is therefore not just one function C : ( V, Π) V −{∅} , but a collection of such functions, onefor each finite set V contained in some countable reference set, say the natural numbers N . Ina similar way, preference aggregation is not defined by a single function F : L ∗ ( V ) → L ( V ) butby a set of such functions, one for each finite V contained in the reference set. However, whenwe define preference aggregation, we usually define it for a fixed V , leaving the dependence on V implicit.Note also that together, Axioms Anonymity and
Embedding imply the isomorphism invariancediscussed in the introduction.
Before we begin to study the properties induced by social choice axioms, we look at the propertiesthat fixed point rules have without any further assumptions. To this end, we will define twoproperties of a community rule C . Property 1 ( Independence of Outside Opinions (IOO)) . A community funcion C satisfies Inde-pendence of Outside Opinions if, for all subsets S ⊆ V and all pairs of preference profiles Π , Π ′ on V such that π ′ s = π s for all s ∈ S , we have that S ∈ C ( V, Π ′ ) ⇐⇒ S ∈ C ( V, Π) . Property
IOO simply states that the preferences of outsiders cannot influence whether ornot a subset is a community. It turns out that this property (and one of our Axioms) is alwayssatisfied by any fixed-point community rule.
Proposition 2.
All fixed-point rules satisfy
Independence of Outside Opinions and
World Com-munity . While we use the embedding axiom to makes statements about subsets of a given ground set V , see, e.g.,Propositions 5 and 7 below, we never use that we can embed a given preference network into an even larger one.Therefore, all results of this paper, except for those involving complexity statements, hold if one restricts oneselfto a finite set V , and only considers preference networks defined on subsets V ⊂ V . roof. Clearly, any fixed point rule satisfies
IOO since the preferences of outsiders are entirelyignored when deciding if a subset constitutes a fixed point. The axiom WC is satisfied vacuously,because it involves looking at all v ∈ V − V .Turning now to social choice axioms, we must first formally define the axioms informallydescribed in Section 1.2. To this end, we need the notion of an election, which will be definedas a triple ( V, F, S ) where V and S are finite sets (called the set of candidates and voters,respectively), and F : L ( V ) ∗ → L ( V ) is a preference aggregation function. Social Choice Axiom 1 ( Unanimity (U)) . An election ( V, F, S ) satisfies Unanimity if, for allpreference profiles, Π S = { π s : s ∈ S } ∈ L ( V ) S and all pairs of candidates, { i, j } ⊆ V , π s ( i ) > π s ( j ) , ∀ s ∈ S = ⇒ F (Π S )( i ) > F (Π S )( j ) . The question then is: what properties capture the intuition behind
Unanimity and how dothey relate to this social choice axiom? To answer this, we define the following two propertiesof a community function C . Property 2 ( Pareto Efficiency (PE)) . A community function, C , is Pareto Efficient if, for a givenpreference network A and a given community S ∈ C ( A ) , it is the case that for all u ∈ S , v / ∈ S ,there is a s ∈ S such that u ≻ π s v . Property 3 ( Clique (Cq)) . A community function C satisfies the Clique
Property if for all A = ( V, Π) , u ≻ π s v, ∀ u, s ∈ S, ∀ v / ∈ S = ⇒ S ∈ C ( A ) . Property
Pareto Efficiency is a negative property that states that subsets in which a non-member is preferred to a member by everyone inside the subset, should not be a community. Incontrast,
Clique is a positive Property, in that it states that a completely self-loving group (i.e.,a clique) must be a community.It turns out that both of these properties are implied by
Unanimity . Proposition 3.
Fix V and a preference aggregation function F , and let C F be the fixed pointrule with respect to F . If all elections ( V, F, S ) with S ( V satisfies Unanimity , then C F satisfiesthe properties Pareto Efficiency and
Clique .Proof.
Fix a preference network A = ( V, Π).First, let us show that C F satisfies Pareto Efficiency . Assume otherwise. In this case theremust be a community S ( V such that for some s ∈ S and j / ∈ S , everyone in S prefers j to s . However, this implies that j must be ranked higher than s in F (Π S ) by Unanimity . By thepigeon hole principle this implies that the elements of S cannot occupy the first | S | positions ofthis preference aggregation, and therefore S is not a community.Now to show that C F satisfies the Clique
Property, assume S ( V is a clique ( ∀ i, j ∈ S and k / ∈ S , j ≻ π i k ). Then all elements of S are preferred by all members of S to all members of V − S and therefore must appear in the first | S | slots of F (Π S ) by Unanimity . This then impliesthat S is a community as required. Social Choice Axiom 2 ( Non-Dictatorship (ND)) . An election ( V, S, F ) is Non-Dictatorial ifthere exists no dictator, i.e., no voter i ∈ S such that F (Π S ) = π i for all preference profiles Π S ∈ L ( V ) S . ND as we did with Unanimity , we do the inverse,and show that a dictatorship violates some of our axioms.
Proposition 4.
Fix V and a preference aggregation function F . If C F , the fixed point rule withrespect to F , satisfies Group Stability or Anonymity , then all elections ( V, F, S ) with S ⊂ V and < | S | < | V | satisfy Non-Dictatorship .Proof.
Assume (
V, F, S ) is dictatorial, with dictator s ∈ S . Let π s be such that all members of S are ranked above those outside of S . Because s is a dictator, we have that S is a community( S ∈ C F ). Additionally let every other member of S rank some non-member v / ∈ S above s .However, if C F satisfies Group Stability , S cannot be a community. Furthermore, if C F satisfies Anonymity , if the preferences of any two members of S are swapped, S should remain acommunity. However, if s swaps with any other member of S , v will be ranked above s in theaggregate preference and thus S cannot be a community.The last of the three social choice axioms, Independence of Irrelevant Alternatives , simplystates that the aggregate relation between any two pairs of candidates should not depend on thepreferences for any other candidate.
Social Choice Axiom 3. ( Independence of Irrelevant Alternatives ) An election ( V, F, S ) satisfies Independence of Irrelevant Alternatives (IIA) if for all preference profiles, Π S , Π ′ S ∈ L ( V ) S andall candidates a, b ∈ V we have that (cid:0) ∀ s ∈ S, a ≻ π s b ⇔ a ≻ π ′ s b (cid:1) = ⇒ (cid:16) a ≻ F (Π S ) b ⇔ a ≻ F (Π ′ S ) b (cid:17) . This axiom is can reasonably be considered the strongest of the three, in that it says thatthe aggregate preference between two candidates does not even depend on the preferences votershave between either of the two and some other candidate. We will demonstrate this strength byproving an impossibility result involving modest assumptions about the fixed point rule of anaggregation function that satisfies
IIA . Theorem 1.
Let F be an aggregation function such that the fixed point rule with respect to F satisfies the Clique
Property and the
Group Stability
Axiom. Then no election ( V, F, S ) with S ⊆ V and < | S | < | V | satisfies IIA .Proof.
Let S ⊆ V such that 1 < | S | < | V | . Assume that the election ( V, F, S ) satisfies
IIA ,and the resulting fixed point rule C F satisfies Cq and GS . We will first show that the election( V, F, S ) must satisfy
Unanimity .In the following preference profiles, Π, Π ′ , Π ′′ ∈ L ( V ) S , we assume that every member of S has the same preference, π , π ′ , and π ′′ respectively. First, let π rank all members of S abovenon-members. By the Clique
Property, S ∈ C F ( A ) and thus ∀ s ∈ S, v / ∈ S, s ≻ F (Π) v. (1)Thus, by IIA , if s ∈ S is unanimously preferred to v / ∈ S , s must be strictly preferred to v in theaggregate preference.Now let π ′ be the same as π only with the least preferred member of S , s ′ , and the mostpreferred non-member, v ′ , switched in rank. By the partial Unanimity property (1), in the12ggregate F (Π ′ ), all members of S − { s ′ } are preferred to all v / ∈ S , and all members of S arepreferred to all v ∈ V − S − { s ′ } . On the other hand, by GS , S / ∈ C F (Π ′ ), which is only possibleis if v ′ (cid:23) F (Π ′ ) s ′ . Applying the partial Unanimity property once more yields the following twostatements: ∀ s ∈ S − { s ′ } , s ≻ F (Π ′ ) s ′ and ∀ v / ∈ S ∪ { v ′ } , v ′ ≻ F (Π ′ ) v, and by IIA , this in turn implies ∀ s ∈ S − { s ′ } , s ≻ F (Π) s ′ and ∀ v / ∈ S ∪ { v ′ } , v ′ ≻ F (Π) v. (2)By IIA , this means that for any two members or two non-members if one is unanimously preferredto the other, then it must be strictly preferred in aggregate preference. Indeed, consider, e.g., s, s ′ ∈ S and a profile ˜Π S such that s ≻ ˜ π i s ′ for all i ∈ S . Choose Π in such a way that everymember has the same profile, s ′ has rank | S | and s ≻ π i s ′ for all i ∈ S . By IIA , s ≻ F ( ˜Π) s ′ ⇐⇒ s ≻ F (Π) s ′ , so by (2), s is preferred to s ′ in aggregate.Finally, consider π ′′ where v ′ is switched with the second lowest ranked member, s ′′ . By theabove additional partial Unanimity property, s ′ must be strictly preferred to s ′′ in the aggregatepreference F (Π ′′ ), and therefore v ′ must be strictly rather than weakly preferred to s ′′ in theaggregate preference. Thus, again by IIA , if a non-member, v / ∈ S , is unanimously preferred toa member s ∈ S , v must be strictly preferred to s in the aggregate preference. Taken together,these three partial Unanimity properties, constitute
Unanimity .Since the election (
V, F, S ) satisfies both
IIA and
Unanimity , by Arrow’s Impossibility Theo-rem [2] it must be a dictatorship, contradicting Proposition 4.
Here we state some additional properties of interest that community rules (not necessarily fixedpoint rules) have when they satisfy one or more of our main axioms.
Proposition 5.
Let C be a community rule that satisfies the World Community and
Embedding
Axioms. Then C must also satisfy the Cliques
Property.Proof.
Let A = ( V, Π) be a preference network and S be a clique (every member of S prefers S to V − S ). By World Community , we have that S ∈ C (( S, Π | S )) and by Embedding we have C ( A ) ∩ S = C (( S, Π | S )). Therefore S is a community. Proposition 6.
Any community rule C that satisfies Monotonicity must satisfy
Independence ofOutside Opinions .Proof.
Let A = ( V, Π) be a preference network. Axiom
Mon features an alternative preferenceprofile Π ′ stating that if Π ′ satisfies certain properties and S is a community for ( V, Π ′ ), then S must be a community ( V, Π). Because the axiom places no restrictions on the preferences ofvoters from V − S , the rule C must satisfy IOO . Property 4 ( Outsider Departure (OD)) . A community rule C satisfies the Outsider Departure
Property if for a given preference network A = ( V, Π) , community S ∈ C ( A ) , and outsider v / ∈ S ,we have that S ∈ C ( V − { v } , Π | V −{ s } ) . roposition 7. A community rule, C , that satisfies the Monotonicity and
Embedding
Axiomsmust also satisfy the
Outsider Departure
Property.Proof.
Let A = ( V, Π) be a preference network, S ∈ C ( A ) a community, and v / ∈ S an outsider.Consider the preference profile Π ′ that ranks v at the end of everyones preference. By Mon , S ∈ C ( V, Π ′ ). Furthermore, since Π ′ satisfies the setup for Embedding , we also have S ∈ C ( V −{ v } , Π ′ | V −{ v } ). However, Π ′ | V −{ v } = Π | V −{ v } since Π and Π ′ only differ in the placement of v .Therefore we have S ∈ C ( V − { v } , Π | V −{ s } ). Proposition 8.
If a community rule satisfies the
Group Stability and
Self-Approval
Axioms itmust satisfy the
Pareto Efficiency
Property.Proof.
Let S be a community.Case 1: | S | = 1. By Self-Approval , the one member s must rank herself above all outsidersand therefore satisfies PE .Case 2: | S | >
1. Choose G ⊂ S such that G is a singleton { s ′ } . By Group Stability , for alloutsider singletons { g ′ } ⊆ V − S and bijections ( f i : { s ′ } → { g ′ } , i ∈ S − G ) there exists an s ∈ S − G such that s ′ ≻ π s f s ( s ′ ). Since it is clear that f s ( s ′ ) = g ′ , s provides the necessarywitness for s ′ and S satisfies PE . We now examine several examples of aggregation based community rules through the lens ofour axiomatic framework. In Section 3.1, we focus on a what we call weighted fixed-pointrules, starting with the B CT community function from [3]. We show that it violates bothAxioms
Monotonicity and
Group Stability . The violation of the monotonicity axiom was initiallysomewhat of a surprise and rather counterintuitive to us. This violation is illustrative of thesubtlety of community rules; indeed, it helped us to identify a weaker monotonicity propertythat the B CT function satisfies. We then show that the fixed-point community rule based onany Borda-count-like voting function is inconsistent with either the
Group Stability axiom orthe
Clique property. This impossibility result and Theorem 1 illustrate some basic limitationsof fixed-point community rules. Next, we study the properties of the harmonious communityfunction in Section 3.2. We will show that it can be obtained by preference aggregation, andthat it obeys all of our axioms except for Axiom GS . It does, however, satisfy a weaker versionof this axiom, see Theorem 4. In our final subsection, Section 3.3, we compare the three rulesBorda voting, B CT voting, and the harmonious rule.
This section focuses on a class of community rules that lie in between general fixed point rulesand the B CT community rule, which we call weighted fixed point rules . First, we will look atsome of the properties of the B CT rule as a particular case of a weighted fixed point rule.
Theorem 2.
The B CT community rule, C B CT , does not satisfy Monotonicity or Group Stabil-ity . It satisfies all other axioms, as well as Properties
Pareto Efficiency and
Clique . roof. Directly from the definition of the B CT voting function φ Π S , C B CT satisfies Axioms A , WC , Emb , and Properties PE and Cq . Suppose C B CT does not satisfy SA . Then, there exists apreference network A = ( V, Π), S ∈ C B CT ( A ), T ⊆ V − S , and a tuple of bijections ( f s : S → T )such that for all s, u ∈ S , u ≺ π s f s ( u ). It follows that ∀ s ∈ S , the numbers of votes cast by s for S according to φ Π S is less than the numbers of votes that s casts for T . Summing up thevotes from S , the average votes that members of T receive is larger than the average votes thatmembers of S receive, contradicting the assumption that everyone in S receives more votes thaneveryone in T . Thus, C B CT satisfies SA .To show C B CT satisfies Axiom CRM , consider S , Π and Π ′ as in Axiom CRM . By thecoherence assumption for members, there exists σ ∈ L ( S ) such that for s , s ∈ S , for all s ∈ S , s ≻ π ′ s s if and only if s ≻ σ s .Let s ∗ denote the least preferred elements of S according to σ . By the assumption that π s ( v ) = π ′ s ( v ) for all s ∈ S, v ∈ V − S , we have that π s ( V − S ) = π ′ S ( V − S ), and hence alsothat π s ( S ) = π ′ s ( S ). But this implies that for all u ∈ Sφ Π S ( u ) = X s ∈ S π s ( u ) ≤| S | ≥ X s ∈ S π s ( S ) ⊆ [1: S ] = X s ∈ S π ′ s ( S ) ⊆ [1: S ] = X s ∈ S π ′ s ( s ∗ ) ≤| S | = φ Π ′ S ( s ∗ ) . If S ∈ C B CT ( V, Π ′ )), then s ∗ receives more votes from Π ′ S than every v ∈ V − S , and the numberof votes v receives from Π S is the same as the number of votes it receives from Π ′ S . On the otherhand, for all u ∈ S , the number of votes u receives from Π S is at least the number of votes s ∗ receives from Π ′ S , implying that S ∈ C B CT ( V, Π). We can similarly show that C B CT satisfiesAxiom CRNM .Let V = [1 : 6], S = [1 : 3], let Π = ( π , ..., π ) be the preference profile π = [142356] , π = [253416] , π = [631425] π = [456123] , π = [156423] , π = [165423]and let Π ′ be the preference profile π ′ = [142356] , π ′ = [234516] , π ′ = [314625] π ′ = π , π ′ = π , π ′ = π . Then S = [1 : 3] ∈ C B CT ( V, Π), as each members of S receives two votes while everyone in [4 : 6]receives only one vote. However, in violation of Axiom Mon , S is no longer a B CT communityw.r.t Π ′ , since 4 now receives three votes, one more than 1, 2 and 3.Note also T = (1 , , ∈ C B CT ( V, Π). Let G = { , } ⊂ T and G ′ = (2 , ⊂ V − T . Asmember 1 prefers 2 to 5 and 4 to 6, T does not satisfy Group Stability .Note that the same analysis shows that C B CT does not satisfy the Outsider Departure
Prop-erty. In the example above, if member 5 leaves the system, then member 4 will receive 2 votesfrom S = { , , } , and hence S is no longer a C B CT -community.Even though C B CT does not satisfy Mon , it does enjoy the following monotonicity property.
Property 5 ( Outsider Respecting Monotonicity) . If S is a community of a preference network A = ( V, Π) , then S remains a community of ( V, Π ′ ) for any Π ′ such that (1) u ≻ π s t ⇒ u ≻ π ′ s t , ∀ u, s ∈ S, t ∈ V , and (2) v ≻ π s v ′ ⇒ v ≻ π ′ s v ′ , ∀ v, v ′ ∈ V − S, s ∈ S .
15e now analyze the fixed point rule defined by the family of aggregation functions, such asBorda count and B CT voting, that derive a cardinal social preference from ordinal individualpreferences.Let W be a sequence of weight vectors w i ∈ R n , W = ( w , w , . . . ), where n is the number ofelements in V . For a non-empty finite S ⊂ N and Π S ∈ L ( V ) S define the aggregate preference F W (Π S ) on V by i ≻ F (Π s ) j ⇐⇒ X s ∈ S w | S | π s ( i ) > X s ∈ S w | S | π s ( j ) . In other words, i ≻ j in the aggregate iff the total weight of the votes i receives from S is largerthan the total weight of the votes j receives from S , where a vote in position p gets weight w | S | p .In B CT, w k is the vector of k ones followed by ( n − k ) zeros , while Borda count uses w k = ( n, n − , ...,
1) for all k . Definition 8 (Weighted Fixed Point Rule) . For a sequence of vectors W = ( w , w , . . . ) in R n , C W is the fixed point rule with respect to F W . Proposition 9.
Weighted fixed-point rules satisfy Axiom
Anonymity . They satisfy
OutsiderRespecting Monotonicity if w ki ≥ w kj for all k ∈ [1 : n − and i ≤ j , and they satisfy the Clique
Property if and only if for all k ∈ [1 : n − the weight vector w k is such that w ki > w kj for i ≤ k and j > k .Proof. The proof of the first two statements and the “if” part of the third follow directly fromthe definitions. To see the “only if” part of the third statement, consider k, i, j such that w ki ≤ w kj , and let S, Π ∈ L ( V ) be such that | S | = k , π s ≤ k for all s, u ∈ S , and π s ( v ) = π t ( v )for all s, t ∈ S, v ∈ V . Then S satisfies the condition of the Property Cq , but it is not acommunity. To see this, choose v ∈ S and v ′ / ∈ S such that π s ( v ) = i and π s ( v ′ ) = j . Then P s ∈ S w kπ s ( v ) ≤ P s ∈ S w kπ s ( v ′ ) , showing that S is not a community.Together with Proposition 5, the next theorem implies that there is no weighted fixed pointrule that satisfies the Group Stability , World Community and
Embedding
Axioms.
Theorem 3. (Impossibility of Weighted Aggregation Schema)
Weighted Fixed PointRules are inconsistent with either the
Group Stability
Axiom or the
Clique
Property.Proof.
Let A = ( V, Π) be a preference network, S ⊂ V , and C W a weighted fixed point rulesatisfying the the Clique
Property. Throughout the the proof, we will take V = { a, b, c, d, e } and S = { a, b, c } , and consider preference profiles such that S violates Group Stability . In order for C W to obeythe Axiom GS , we would need the weight vector w ∈ R to be such that S / ∈ C ( V, Π) for allΠ considered in this proof. Our goal is to show that this will lead to a contradiction. We startunder the assumption that the weights are decreasing, i.e., in addition to the already established The rule C B CT does not specify what the weight w k should be for k > n since preferences with more votersthan alternatives do not occur when determining communities – so we are free to define it arbitrarily, say w ki = 1for all i if k > n . w i > w j when i = 1 , , j = 4 , C W satisfies the the Clique
Property), wewill first assume that w ≥ w ≥ w and w ≥ w .Consider the following scenario: π a = [ adebc ] , π b = π c = [ abcde ] . Since a prefers d and e over b and c , S is not group stable and hence cannot be a community. Byour assumption that w ≥ w ≥ w > w ≥ w , we have that a ≻ F W (Π S ) b (cid:23) F W (Π s ) c ≻ F W (Π S ) e and b ≻ F W (Π S ) d . Therefore the only way S cannot be a community is that d (cid:23) F W (Π S ) c , i.e., w + 2 w ≥ w + w . Notice that this implies that we cannot have both w = w and w = w .Now consider a modified preference profile: π ′ a = π ′ b = [ abdce ] , π ′ c = [ caebd ] . In this profile a and b prefer d over c , so again S violates GS and hence cannot be a community.On the other hand, we now have a ≻ F W (Π ′ S ) b , b (cid:23) F W (Π ′ S ) d ≻ F W (Π ′ S ) e . Thus we must haveeither b ∼ F W (Π ′ ) d or d (cid:23) F W (Π ′ S ) c . The former, however, implies w = w and w = w and ishence a contradiction. Therefore the latter must be true which implies2 w + w ≥ w + 2 w . This brings us to the final preference profile: π ′′ a = [ abdce ] , π ′′ b = [ dcabe ] , π ′′ c = [ cbaed ] . Again a and b prefer d to c , so the profile violates GS , and hence again can’t be a community.Now a ≻ F W (Π ′′ ) c (cid:23) F W (Π ′′ ) b and d ≻ F W (Π ′′ ) e , showing that for S not to be a community, wemust have d (cid:23) F W (Π ′′ ) b , which gives w + w + w ≥ w + w . Defining d i = w i − w i − , we can write the bounds obtained so far as d ≤ d + d d + d + d ≤ d d + d + d ≤ d . Chaining up these three bounds, we get d + d ≥ d ≥ d + d + d ≥ d + d + d + d + d = 2( d + d ) + d , contradicting our assumption d i ≥ Cq implies d > S still violates GS . In other words, for any permutation σ of [1 : 5] thatleaves [1 : 3] and [4 : 5] invariant, S violates GS under the profiles { σ ◦ π s } s ∈ S , { σ ′ ◦ π s } s ∈ S , and { σ ′′ ◦ π s } s ∈ S . Choosing the permutation in such a way that the weights ˜ w i = w σ ( i ) are ordered,we obtain the above three inequalities for the weights ˜ w i , leading again to a contradiction.17 .2 Properties of Harmonious Communities In this subsection, we analyze the harmonious community function given by Definition 5. Wefirst prove that it can be expressed in terms of a suitable preference aggregation function.
Proposition 10.
There exists a preference aggregation function F H : L ( V ) ∗ → L ( V ) such thatthe harmonious community function H is defined by a F H .Proof. Given V , a finite set S , and a preference profile Π S ∈ L ( V ) S , we consider the followingdirected graph G Π S = ( V, E Π S ) where ( i, j ) ∈ E Π S if at least half of S prefers i to j . Note thatif | S | is an odd number, then G Π S is a tournament graph. If | S | is an even number, then E Π S contains both ( i, j ) and ( j, i ) if exactly half of Π S prefer i to j . G Π S is total since for all i, j ∈ V ,either ( i, j ) ∈ E Π S or ( j, i ) ∈ E Π S . Because G Π S is total, the graph ˆ G Π s obtained from G Π S bycontracting each strongly connected component into a single vertex is an acyclic , tournamentgraph. As a consequence, the graph ˆ G Π s has exactly one Hamiltonian path that totally ordersits vertices. Let ( V , ..., V t ) be the strongly connected components of G Π S , sorted by the orderdetermined by the Hamiltonian path. The partition ( V , ..., V t ) of V then defines an orderedpartition F H (Π S ), with V i ≻ F H (Π S ) V j iff i ≤ j .Next, we consider a subset T ⊂ V . It is then easy to check that if T is of the form T = ∪ j ≤ i V j for some i ∈ [1 : t ], then for all u ∈ T, v ∈ V − T , a majority of S prefers u to v , and vice versa.Specializing to S = T , we see that H is defined by the preference aggregation function F H .Next we show that H satisfies all axioms except for Group Stability . Theorem 4.
The harmonious community function satisfies Axioms A , SA , Mon , Emb , WC , CRM , and
CRNM , but it does not satisfy GS .Proof. Directly from the definitions, one easily checks that H satisfies Axioms A , Mon , Emb and WC .By a similar argument to the proof of Theorem 2, we can prove that H satisfies SA : if S ∈ H ( A ) does not satisfy SA , then there exists a T ⊂ V − S of the same size as S such thateach s ∈ S lexicographically prefers T over S . With the help of Proposition 1, this implies that,for each s ∈ S , there are at least (1 + 2 + · · · + | S | ) pairs ( u, v ) ∈ S × T such that s prefers v over u . Thus the number of triples ( s, u, v ) such that s ∈ S prefers v ∈ T over u ∈ S is at least | S | ( | S | + 1) /
2. However, S ∈ H ( A ) implies that this number has to be strictly smaller than | S | / H is consistent with Axiom CRNM , consider a preference profile Π , Π ′ as specifiedin Axiom CRNM . By the coherence assumption on non-members, there exists a linear order σ on V − S , such that ∀ i, j ∈ V − S and ∀ s ∈ S , i ≻ π ′ s j ⇔ i ≻ σ j . Let v ∗ be the most preferredelement of σ . By the assumption that π s ( u ) = π ′ s ( u ) for all s, u ∈ S , we have π s ( S ) = π ′ s ( S )and hence also π s ( V − S ) = π ′ s ( V − S ). But this implies that for all v ∈ V − S , π s ( v ) ≥ min { i ∈ π s ( V − S ) } = min { i ∈ π ′ s ( V − S ) } = π ′ s ( v ∗ ) . We therefore have shown that for all s, u ∈ S such that u ≻ π ′ s v ∗ , we have that u ≻ π s v for all v ∈ V − S . Assume now that S ∈ H (( V, Π ′ )). Then for all u ∈ S , the majority of (Π ′ , S ) prefer u to v ∗ , which, as we just have shown, implies that for all v ∈ V − S , the majority of (Π , S )18refer u to v , which in turn implies that S ∈ H (( V, Π)). We can similarly show that H satisfiesAxiom CRM .The set T in the proof of Theorem 2 is also an example that H violates Axiom GS .While H does not satisfy the GS Axiom, it satisfies the following weaker property.
Property 6.
Weak Group Stability
For all preference profiles Π on V and all S ∈ C ( V, Π) , S is weakly group stable . Here a set S ⊂ V is called weakly group stable if for all G ⊂ S , G ′ ⊂ V − S s.t. < | G | = | G ′ | ≤ | S | / , and all bijections ( f : G → G ′ , i ∈ S − G ) there exists s ∈ S − G , u ∈ G such that u ≻ π s f ( u ) . Note that the property is weaker than the GS Axiom in two ways: we restrict ourselves togroups G of size at most | S | /
2, and we only allow for a global bijection f , rather than individualbijections f s . Proposition 11. H is weakly group stable , while the Borda count and the B CT rule are not.Proof.
Consider a set S ∈ H ( V, Π), subsets G ⊂ S and G ′ ⊂ V − S such that 0 < | G | = | G ′ | ≤| S | /
2, and a bijection f : G → G ′ . For each u ∈ G the majority of S prefer u to f ( u ) (who is nota member of S ), and since | G | ≤ | S | /
2, this implies that there must be at least one s ∈ S − G such that s prefers u to f ( u ), as required.To give a counterexample for both Borda counting and the B CT rule, consider V = [1 : 6], G = [3 : 4] and G ′ = [5 : 6], with preference profiles π = [125463] , π = [126354] , π = [341256] , π = [341256] . Then 1 and 2 prefer 5 over 4, and 6 over 3, but S is a community both with respect to B CT(where 1 and 2 get four votes, 3 and 4 get three votes, and 5 and 6 get only one vote), and withrespect to Borda count (with counts 20 , , , , , , . . . ,
6, respectively).
Proposition 12. H satisfies IOO as well as Cq and the PE , but F H does not satisfy U .Proof. By Proposition 2, H satisfies IOO . To see that it does not satisfy U , let V = { a, b, c } ,let S = { a, b } and π a = ( acb ), π b = ( bac ). Then a ≻ π s c for all s ∈ S , and both a ≻ π s b and b ≻ π s c in half of S . Therefore ( ac ) , ( cb ) , ( bc ) , ( ab ) , ( ba ) ∈ E Π S . Thus, a, b, c belongs to thesame connected component in G Π S , showing that S is not a harmonious community. To seethat H satisfies both Cq and PE in spite of the fact that it does not satisfy the assumptions ofProposition 3, we use Proposition 5 to infer Cq , and the observation that S ∈ H ( A ) implies thatfor any a pair of elements ( u ∈ S, v S ), the majority of S prefer u over v , proving PE . CT voting, and the harmonious rule
In this subsection, we compare the fixed-point community rules that we have discussed so far:Borda voting, B CT voting, and the harmonious rule. While all three have their own appealingsimplicity and intuition and all satisfy Axioms A , SA , Emb , WC , CRM , and
CRNM , there aresignificant differences with respect to Axioms
Mon and GS , and the Outsider Departure property.19
Outsider Departure : A harmonious community S remains a harmonious community whenany outsider v S leaves the system since the departure does not alter any pairwisepreferences. However, for a B CT community S , the departure of an outsider can increasethe votes for other outsiders enough to destabilize the B CT community. In a similar way,one can see that the Borda count rule is also unstable to departure of an outsider. • Monotonicity : The harmonious rule satisfies Axiom
Mon . The other two only satisfy theweaker
Outsider Respecting Monotonicity property . • Group Stability : The subset T in the proof of Theorem 2 is a community according to allthese three community rules. But T violates GS because 1 prefers outsiders over 5 and 6,even though 5 and 6 prefer 1 over everyone else: Element 1 is an “arrogant” member of itscommunity. All aggregation functions satisfying Unanimity seem to be prone to existenceof “arrogant” members. The harmonious rule satisfies the stability of majority subgroupunder a global bijection f , although the stability of the minority subgroup (or the majoritysubgroup with individual bijections f s ) may not be guaranteed. The fixed-point rule ofBorda count and B CT voting essentially have no guarantee of group stability. • Small World : In general, we say a community function C satisfies the Small World propertyif S ∈ C (( V, Π)) ⇔ ∀ U ⊆ V − S, | U | < | S | , S ∈ C ( S ∪ U, Π | S ∪ U ) . This Helly-type property [5] localizes the identification of a community. Note that the
Small World property includes some form of
Outsider Departure together with the propertythat every community is “locally” verifiable. One can easily show that the fixed-pointrules of the Borda count or B CT voting do not have the
Small World property, while theharmonious rule enjoys the following stronger variant of the small world property S ∈ H (( V, Π)) ⇔ ∀ v ∈ V − S, S ∈ H ( S ∪ { v } , Π | S ∪{ v } ) , and hence the property given in (3). In this section, we characterize the taxonomy of the axiom-conforming community rules.First, in Section 4.1, we define two rules, the Clique Rule and the Comprehensive Rule,which satisfy all axioms, and which are most selective and most comprehensive, respectively, inthe sense that any rule which satisfies all axioms leads to a set of communities which containsall communities defined by the Clique Rule and is contained in the Comprehensive Rule (thestatement that this is the case, Theorem 5, will be our main theorem in this subsection).In the next subsection, Section 4.2, we then expand on this “Taxonomy Theorem”, and showthat under the following natural intersection and union of community rules, the family of allcommunity rules that satisfies all eight axioms forms a bounded lattice. We will use the followingtwo set-theoretic operators of community functions to define these lattice structures. Again, we can use the profiles from the proof of Theorem 2 to show that the Borda count rule does not satisfy
Mon . efinition 9 (Operations over Community Rules) . For two community functions C and C ,we define the intersection and union, C ∩ C and C ∪ C , as the community functions which,for all preference networks A , respectively satisfy ( C ∩ C )( A ) := C ( A ) ∩ C ( A )( C ∪ C )( A ) := C ( A ) ∪ C ( A ) . We start with perhaps the simplest rule for communities that satisfies the
Clique
Property.
Rule 1 (Clique Rule ( C clique )) . A non-empty subset S ⊆ V is a community of A = ( V, Π) , ifand only if ∀ u, s ∈ S , v / ∈ S , u ≻ π s v . We use C clique to denote the community function definedby this rule. Proposition 13. C clique satisfies all Axioms.Proof. The (easy) proof is left as an exercise for the reader.However, the clique rule appears to be too restrictive, since it has the following structuralfeature, which essentially rules out any non-trivial overlap of communities, while “Real-world”communities typically have non-trivial overlaps among themselves.
Proposition 14.
For any preference network A , if S , S ∈ C clique ( A ) , then either S ∩ S = ∅ or S ⊂ S , or S ⊂ S .Proof. Assume otherwise. By assumption, we can choose an element s ∈ S ∩ S . Without loss ofgenerality assume | S | ≤ | S | . Again by assumption, there exists an element s ′ ∈ S and s ′ / ∈ S .By the definition of the C clique s must have s ′ in its top | S | choices. However, this means that s ′ is also in the top | S | choices for s , which violates the fact that S is in C clique ( A ).Next we address the question of whether there are rules consistent with all axioms that admitoverlapping communities. To address this question, we consider rules defined by communityaxioms. Rule 2 (Axiom Based Community Rules) . For X ∈ { GS, SA } let C X be the community ruledefined by A = ( V, Π)
7→ C X ( A ) , where C X ( A ) is the set of non-empty subsets S ⊂ V such that S obeys axiom X . For example, C GS denote the community rule that S ∈ C GS ( A ) if and only S enjoys the Group Stability
Axiom.The first part of our Taxonomy Theorem is a direct consequence of the following basic lemma.
Lemma 1. (Intersection Lemma: GS and SA ) For X ∈ { A, Mon, CRM, CRNM, WC, Emb } ,if C satisfies Axiom X , then e C = C ∩ C GS ∩ C SA satisfies Axioms X , GA and SA .Proof. C GS and C SA are both consistent with A , WC , and Emb , thus if C satisfies Axiom X ∈{ A, WC, Emb } , then e C remains consistent with Axiom X.To see e C satisfies Axiom Mon if C satisfies Mon , choose Π , Π ′ such that, for all u, s ∈ S and v ∈ V , u ≻ π ′ s v = ⇒ u ≻ π s v . We need to show that if S ∈ e C (( V, Π ′ )) then S ∈ e C (( V, Π)).21uppose this is not the case, then either (1) S
6∈ C GS (( V, Π)) or (2) S
6∈ C SA (( V, Π)). In Case(1), there exists G ⊂ S , G ′ ⊂ V − S , | G | = | G ′ | , and bijections ( f s : S → G ′ | s ∈ S − G ) such that ∀ s ∈ S − G, ∀ u ∈ G , u ≺ π s f s ( u ). Then by the condition stated in Mon , we have u ≺ π ′ s f s ( u ),which shows S
6∈ C GS ( A ′ ). In Case (2), there exists G ′ ⊂ V − S , bijections ( f s : S → G ′ ) suchthat ∀ s, u ∈ S , u ≺ π s f s ( u ). Then by the condition stated in Mon , we have u ≺ π ′ s f s ( u ), whichimplies that S
6∈ C GS ( A ′ ).Suppose C satisfies Axiom CRM . Consider Π , Π ′ as specified in Axiom CRM . Given s ∈ S ,the profiles π s and π ′ s are then assumed to be identical on V − S , implying in particular that π s ( V − S ) = π ′ ( V − S ), and hence also that π s ( S ) = π ′ s ( S ). Furthermore, by the coherenceassumption for members, there exist σ ∈ L ( S ) such that ∀ u , u , s ∈ S , u ≻ π ′ s u iff u ≻ σ u .We need to show that if S ∈ e C (( V, Π ′ )) then S ∈ e C ( A ). Suppose this is not the case, then either(1) S
6∈ C GS ( A ) or (2) S
6∈ C SA ( A ).In Case (1), there exists G ⊂ S , G ′ ⊂ V − S , | G | = | G ′ | , a set of bijections ( f s : G → G ′ , s ∈ S − G ), such that ∀ s ∈ S − G, u ∈ G , u ≺ π s f s ( u ). Let T ⊂ S be the set of | G | leastpreferred elements by σ . We now show that there exists bijections ( f ′ s : T → G ′ , s ∈ S ) suchthat ∀ s ∈ S − T, u ∈ T , u ≺ π ′ s f ′ s ( u ), which would imply that S
6∈ C GS (( V, Π ′ )).Let us denote T by T = { t , ..., t | T | } such that t i ≺ σ t i +1 . Fix an s ∈ S − T , and let usdenote G by G = { g , ..., g | T | } such that g i ≺ π s g i +1 , and denote G ′ by G ′ = { g ′ , ..., g ′| T | } suchthat g ′ i ≺ π s g ′ i +1 . By Proposition 1, we then have that g i ≺ π s g ′ i for all i = 1 , . . . , | T | . Inother words, π s ( g i ) > π s ( g ′ i ). We define f ′ s by mapping t i to g ′ i . Note that the positions ofthe preferences rankings of S as a set are the same in π ′ s and π s . Because T is the set of | G | least preferred elements of S , we have π ′ s ( t i ) > π s ( g i ). Since π ′ s ( g ′ i ) = π s ( g ′ i ) it then follows that π ′ s ( t i ) > π s ( g i ) > π s ( g ′ i ) = π s ( g ′ i ). Thus, t i ≺ π ′ s g ′ i , and consequently, S
6∈ C GS (( V, Π ′ )). In Case(2), there exists G ′ ⊂ V − S and a set of bijections ( f s : S → G ′ , s ∈ S ), such that ∀ s, u ∈ S , u ≺ π s f s ( u ). By the similar argument as in Case (1) (by setting T = S ), we can show that thereexists bijections ( f ′ s : S → G ′ , s ∈ S ) such that ∀ s ∈ S, u ∈ S , u ≺ π ′ s f ′ s ( u ), which implies that S
6∈ C GS (( V, Π ′ )). Thus, e C satisfies Axiom CRM .We can similarly prove that C satisfies CRNM if C satisfies it.Finally, by definition, C ∩ C GS ∩ C SA satisfies GS and SA . Rule 3. (Comprehensive Community Rule)
For a preference network A = ( V, Π) , a non-empty S ⊆ V is a community according to C comprehensive if and only if S satisfies both GroupStability and
Self-Approval axioms. In other words, C comprehensive := C GS ∩ C SA . We now prove that C comprehensive is indeed the most comprehesive community rule thatsatisfies all Axioms. Theorem 5 (Taxonomy: Lattice Top and Bottom) . C comprehensive satisfies all Axioms. More-over, for any community function C that satisfies all Axioms, for every preference network A = ( V, Π) C clique ( A ) ⊆ C ( A ) ⊆ C comprehensive ( A ) . (3) Proof. C all ( A ) = 2 V −{∅} satisfies Axioms A , Mon , CRM, CRNM , WC and Emb . Since C comprehensive = C all ∩ C GS ∩ C SA , by the Intersection Lemma, C comprehensive satisfies all Axioms.22n the other hand, by Proposition 5, any rule which satisfies WC and Emb , must satisfy the
Cliques
Property, so for any C that satisfies all axioms, C clique ( A ) ⊆C ( A ) ⊆ C GS ( A ) ∩ C SA ( A ).Thus C clique ( A ) ⊆ C ( A ) ⊆ C comprehensive ( A ) . Theorem 5 shows that C comprehensive and C clique are the most inclusive and the most selectivefunction, respectively, that satisfies all axioms. While it is very easy to determine whethera subset in a preference network satisfies Property Clique , in Section 5 we demonstrate that C comprehensive is highly “non-constructive” by showing that the decision problem for determiningwhether a subset in a preference network satisfies Axiom Self-Approval or Group Stability is coNP-complete . The Intersection Lemma provides us with a tool for exploring the taxonomy of community rules.In this subsection, we continue this exploration and make it more systematic using two latticestructures enjoyed by the community-rule taxonomy.
Theorem 6 (Taxonomy: Lattice Structures of Community Rules) . Let C denote the family ofall community rules that satisfies all eight axioms. Let C B be a superset of C that denotes thefamily of all community rules that satisfies Axioms A, Mon, CRM, CRNM, WC, Emb .1. The algebraic structure T = ( C , ∪ , ∩ , C clique , C comprehensive ) forms a bounded lattice with C clique as the lattice’s bottom and C comprehensive as the lattice’s top.2. The algebraic structure T B = ( C B , ∪ , ∩ , C clique , C all ) forms a bounded lattice with C clique asthe lattice’s bottom and C all as the lattice’s top.Proof. First, by definition, the two operations ∩ and ∪ over the community functions are bothcommunitative and associative. One can easily show that the two operations ∩ and ∪ satisfythe absorption property , that is, for any two C , C ∈ C C ∪ ( C ∩ C ) = C . C ∩ ( C ∪ C ) = C . For example, to see the first one, for any affinity network A , we have( C ∪ ( C ∩ C ))( A ) = C ( A ) ∪ ( C ∩ C )( A ) = C ( A ) ∪ ( C ( A ) ∩ C ( A )) = C ( A ) . To complete the proof that T and T B are lattices, we need to prove that T and T B are closedunder ∩ and ∪ . We organize the arguments as following: • A, WC : it is obvious that if C and C satisfies Axioms A and WC then both C ∪ C and C ∩ C also satisfies Axioms A, WC . • Mon, CRM, CRNM : Suppose A = ( V, Π), A ′ = ( V, Π ′ ), and S ⊂ V are, respectively,two preference networks and a set considered in Axiom Mon . Then if C and C satisfy Mon , we have S ∈ C i ( A ′ ) ⇒ S ∈ C i ( A ) for i ∈ ,
2. Thus, if S ∈ C ( A ′ ) ∩ C ( A ′ ) then S ∈ C ( A ) ∩ C ( A ), and if S ∈ C ( A ′ ) ∪ C ( A ′ ) then S ∈ C ( A ) ∪ C ( A ). Thus, both C ∪ C and C ∩ C also satisfy Axioms Mon . We can argue analogously for Axioms
CRM and
CRNM . 23
Emb : If both C and C satisfy Emb , then for any A = ( V, Π) and any “embedded world” A ′ = ( V ′ , Π ′ ) such that Π , Π ′ satisfy the assumption of Axiom Emb , we have C i ( A ′ ) = C i ( A ) ∩ V ′ for i ∈ { , } . So C ( A ′ ) ∩ C ( A ′ ) = (cid:16) C ( A ) ∩ V ′ (cid:17) ∩ (cid:16) C ( A ) ∩ V ′ (cid:17) = ( C ( A ) ∩ C ( A )) ∩ V ′ C ( A ′ ) ∪ C ( A ′ ) = (cid:16) C ( A ) ∩ V ′ (cid:17) ∪ (cid:16) C ( A ) ∩ V ′ (cid:17) = ( C ( A ) ∪ C ( A )) ∩ V ′ . Thus, both C ∪ C and C ∩ C also satisfies Axioms Emb .Together, this shows that ∀C , C ∈ C B , C ∩ C ∈ C B and C ∪ C ∈ C B . Thus, T B =( C B , ∪ , ∩ , C clique , C all ) is a lattice with C all as the lattice’s top and C clique as the lattice’s bottom(where the former follows from the fact that C all ( A ) satisfies Axioms A , Mon , CRM, CRNM , WC and Emb , while the latter follows from Proposition 5). • GS, SA : Assume C ∈ C and C ∈ C satisfy Axioms GS and SA . We can then argue as forAxiom Mon above to show that both C ∪ C and C ∩ C satisfies Axioms GS, SA .Thus, T = ( C , ∪ , ∩ ) is a lattice. By Theorem 5, C comprehensive is the lattice’s top and C clique asthe lattice’s bottom of T .Theorem 6 allows us to have a notion of the closure of an arbitrary community rule withrespect to these six axioms. In order to define it, we say that a community rule C c ontains arule C if C ( A ) ⊂ C ( A ) for all preference networks A . Theorem 7.
Given a community rule C , there exists a unique smallest community rule, denoted C , that contains C and satisfies all community axioms besides SA and GS .Proof. Consider the set b C of all community rules that contain C and satisfy these six axioms.Note that it is non-empty because C all is guaranteed to contain C . Apart from some technicalissues to be addressed below, if we take the intersection of all the communities in this set, theresulting rule C will still satisfy all six axioms by the proof of Theorem 6, and thus be thesmallest community rule of the set.The technical issues to which we alluded above stem from the fact that, in general, the set b C contains uncountably many community rules. The community rule C is thus defined by anuncountable intersection, while Theorem 6 a priori only allows one to argue about countablymany intersections. But it turns out that while b C is uncountable, when checking the axioms,one never has to consider more than a finite set of rules, allowing one to apply the reasoningfrom the proof of Theorem 6 to show that C does satisfy all desired axioms.To make this precise, we recall that a community rule is given by a sequence of functions, C V : ( V, Π)
7→ C V ( V, Π) ⊂ V −∅ , where V runs over the non-empty finite subsets of countablereference set V . Expressing both C and the rules in C ′ ∈ b C as sequences, C = ( C V ) and C ′ = ( C ′ V ),we have C V (( V, Π)) = \ C ′ ∈ b C C ′ V (( V, Π)) . Hoverer, when verifying the six axioms for C , we only have to deal with a given finite set V at atime (or, in the case of Axiom Emb , all subsets V ′ ⊂ V of a finite set V ); and for a finite set V , C V can be expressed as the intersection over a finite subset of b C , which means when checkingthe axioms for C V , we can use Theorem 6. 24he Intersection lemma serves as a bridge between the two lattices from Theorem 6: We canobtain the lattice T = ( C , ∪ , ∩ , C clique , C comprehensive ) from the lattice T B = ( C B , ∪ , ∩ , C clique , C all )by intersecting the community functions on the lattice points of T B with C GS ∩ C SA , followed bymerging the lattices points with identical community functions. By moving the intersection upthe lattice T B , we can define more inclusive community rules that satisfy all eight axioms. Forexample, by intersecting the lattice top ( C all ) of T B with C GS ∩ C SA , we obtain the lattice top( C comprehensive ) of T . Remark 1.
Note that Theorem 7 and the Intersection Lemma give us a reasonable mappingfrom arbitrary community rules to community rules that satisfy all our axioms. Namely, fora given community rule, C , first take the unique smallest community rule that contains C andsatisfies all axioms besides SA and GS (as in Theorem 7), then apply the intersection from theIntersection Lemma. The mapping can therefore be formulated as C 7−→ C∩C SA ∩C GS . As an example, consider the community rule C that admits all singletons (i.e., subsets ofsize 1) as communities and nothing else. Because C only violates WC of the axioms besides SA , C in addition to all singletons also contains all cliques (thanks to the influence of Emb ).From this, all communities that don’t satisfy SA are removed: i.e., all singletons that do notrank themselves first. As the reader may have already guessed, what remains happens to be theClique Rule. In other words, C ∩C SA ∩C GS = C clique In a small step up the lattice T B from the Clique Rule, we consider the following communityfunction. Rule 4 (Relaxed Clique Rule) . For a non-negative function g : N → N ∪ { } , a non-emptysubset S ⊆ V is a community in A = ( V, Π) if and only if ∀ u, s ∈ S , π s ( u ) ∈ [1 : | S | + g ( | S | )] .We denote this community function by C Clique ( g ) . Proposition 15. C Clique ( g ) ∈ C B and hence ( C Clique ( g ) ∩ C GS ∩ C SA ) satisfies all eight axioms.Proof. The (straightforward) proof is left to the reader.We will show in Section 5 below that C comprehensive is highly “non-constructive” by provingthat the decision problem for determining whether a subset in a preference network satisfiesAxiom Self-Approval or Group Stability is coNP-complete . On the other hand, we will see thatthe community rule given by ( C Clique ( g ) ∩ C GS ∩ C SA ) can be constructive if g is small, seeProposition 17 in Section 5 below.As g varies from 0 to ∞ , the community function C Clique ( g ) moves up the lattice T B from C clique to C all . The intersection with C SA ∩ C GS provides us a “vertical” glimpse of the taxonomylattice T . In particular, as the community rules along this vertical path become more inclusive(when g increases), they become less constructive for community identification. An alternative“vertical” glimpse can be gained by following “harmonious-path” in the lattice T B for communityrules formulated by pairwise comparisons. Rule 5 (Harmonious Path) . For λ ∈ [0 : 1] , a non-empty subset S is a λ -harmonious community in A = ( V, Π) if ∀ u ∈ S, v ∈ V − S , at least λ -fraction of { π s : s ∈ S } prefer u over v . Wedenote this community function by H λ . Proposition 16.
For all λ ∈ [0 : 1] , H λ ∈ C B . Thus, H λ ∩ C GS ∩ C SA satisfies all eight axioms, ∀ λ ∈ [0 : 1] . Further, for λ ∈ (1 / , H λ satisfies Axiom SA , and therefore all axioms butAxiom GS .Proof. The (easy) proof is left to the reader.Therefore, as λ varies from 1 to 0, the community function H λ moves up the lattice T B from H = C clique to H = C all , and so does its non-constructiveness, see Proposition 18 in Section 5. Group Stability and
Self Approval
In this section, we demonstrate that C comprehensive is highly “non-constructive” by showing thatthe decision problem for determining whether a subset in a preference network satisfies Axiom Self-Approval or Group Stability is coNP-complete . Our reduction also provides examples ofpreference networks derived from instances. Theorem 8.
It is coNP-complete to determine whether a subset S ⊂ V is self-approving in agiven preference network A = ( V, Π) . Before starting the proof, we introduce a notation which we will use throughout this section.Given a preference profile ( V, Π) and a non-empty set S ⊂ V , we say that a set G ′ ⊂ V − S is a witness that S is not self-approving , if S lexicographically prefers G ′ to S , and we say that a pair( G, G ′ ) ⊂ S × ( V − S ) is a witness that S is not group-stable if S − G lexicographically prefers G ′ to G . Finally, we say that G ⊂ S threatens the stability of S if there exists a G ′ ⊂ V − S such that S − G lexicographically prefers G ′ to G . Proof.
We reduce to this decision problem: Suppose c = ( c , . . . , c m ) is a instancewith Boolean variables x = ( x , . . . , x n ) (i.e., c j = { u j , v j , w j } ⊂ ∪ ni =1 { x i , ¯ x i } ). We define apreference network as follows: • V = A ∪ B ∪ D ∪ X has m + n + m +2 n members, where A = { a , . . . , a m } , B = { b , . . . , b n } , D = { d , . . . , d m } , and X = { x , . . . , x n , ¯ x , . . . , ¯ x n } . The distinguished subset will be S = A ∪ B , and for convenience we will denote its complement as U = D ∪ X . • Since we will focus on subset S , here we only define the preferences of members in S . Thepreferences of U can be chosen arbitrarily. – Member b i has preference D ≻ A ≻ { x i , ¯ x i } ≻ { b i } ≻ X − { x i , ¯ x i } ≻ B − { b i } , wherepreferences between elements of each set can be chosen arbitrarily. – Member a j has preference c j ≻ { a j } ≻ D ∪ X − c j ≻ B ∪ A − { a j } , where againpreferences between elements of each set are arbitrary.26ntuitively, members of A are used to enforce clause consistency (i.e., make sure each clauseis satisfied) and members of B are used to enforce variable consistency (no variable to both trueand false at the same time). Subsets of X naturally constitute an assignment of the variables,and D provides necessary padding in order to apply Self-Approval .We now show that S is not self-approving if and only if the instance is satisfiable.In one direction, suppose Y = { y , . . . , y n } where y i ∈ { x , ¯ x i } is a satisfying assignmentfor the instance. Let G ′ = Y ∪ D . Now consider the bijection, f , where f ( a j ) = d j and f ( b i ) = y i . It is not hard to see that for all s ∈ S and all i , f ( s ) ≻ πb i s . All that is left is to findsimilar bijections for each a j . First, note that for a j all bijections f j trivially satisfy f j ( s ) ≻ πa j s where s ∈ B ∪ A − { a j } , since this set is ranked at the bottom of π a j . Therefore it is sufficientto show that there exists an element of G ′ that a j prefers to itself. This happens so long asone of the literals from its clause is in G ′ , which must be true by the fact that Y is a satisfyingassignment.In the other direction, suppose G ′ ⊂ U = D ∪ X is a witness that S is not self-approving . Wenote the following: • D ⊂ G ′ otherwise any b i will have a member of A that cannot be mapped to a morepreferred member of G ′ . • Let Y = X ∩ G ′ . Then | Y | = n by the above fact and the fact that | G ′ | = n + m . • { x i , ¯ x i } ∩ G ′ = ∅ by b i ’s preference, and by the pigeonhole principle the literals of Y areconsistent (i.e. { x i , ¯ x i } * Y ). • c j ∩ Y = ∅ by a j ’s preferences.Therefore the variable assignment implied by Y is a satisfying assignment for the instance.The following “padding” lemma allows us to reduce various complexity results concerningcommunity axioms to Theorem 8. Lemma 2.
Let ∅ 6 = S ⊂ V ⊂ V ′ be such that the size of ˜ S = V ′ − V is at least | S | , and let S ′ = S ∪ ˜ S . Then each preference profile Π on V can be mapped onto a preference profile Π ′ on V ′ such that(i) S ′ ∈ C GS ( V ′ , Π ′ ) ∩ C SA ( V ′ , Π ′ ) ⇔ S ′ ∈ C GS ( V ′ , Π ′ ) .(ii) S ′ ∈ C GS ( V ′ , Π ′ ) ⇔ S ∈ C SA ( V, Π) .Proof. Since | ˜ S | ≥ | S | , we can find a surjective map g : ˜ S → S . Given such a map, define Π ′ arbitrarily, except for the following two constraints: • If s ∈ S , then π ′ s ranks all of S ′ = ˜ S ∪ S before anyone in V − S = V ′ − S ′ ; • If ˜ s ∈ ˜ S , then π ′ ˜ s ranks all of ˜ S first, and then gives the rank π ′ ˜ s ( v ) = | ˜ S | + π g (˜ s ) ( v ) to every v ∈ V = V ′ − ˜ S . 27ince every s ∈ S ⊂ S ′ ranks all of S ′ before V ′ − S ′ , no subset G ′ ⊂ V ′ − S ′ can be lexico-graphically preferred by π ′ s to a subset of S ′ . As a consequence, S ′ is trivially self-approvingwith respect to Π ′ , proving statement (i).Furthermore, G cannot threaten the stability of S ′ if G ⊂ S ′ is such that ( S ′ − G ) ∩ S = ∅ . If G ⊂ S ′ threatens the stability of S ′ , we therefore must have that G ⊃ S . On the other hand, if G ) S , then G contains an element ˜ s ∈ ˜ S which means that no set G ′ can be lexicographicallypreferred G , since all elements of S ′ prefer all of ˜ S to anyone in V ′ − S ′ .Thus G can only threaten the stability of S ′ if G = S . In other words, S / ∈ C GS (Π ′ ) if andonly if there exists G ′ ⊂ V ′ − S ′ such that for all ˜ s ∈ ˜ S = S ′ − G , G ′ is lexicographically preferredto S with respect to π ′ ˜ s = π g (˜ s ) . Since by assumption, the image of ˜ S under g is all of S , this isequivalent to the statement that for all s ∈ S , G ′ is lexicographically preferred to S with respectto π s , which is the condition that G ′ is a witness to S / ∈ C SA (Π), proving statement (ii).Given this lemma, the next two theorems are immediate corollaries to Theorem 8. Theorem 9.
It is coNP-complete to determine whether a subset S ⊂ V is group-stable in agiven preference network A = ( V, Π) . Theorem 10.
It is coNP-complete to determine whether a subset S ⊂ V is a member of C comprehensive = C GS ∩ C SA for a given preference network A = ( V, Π) . C Clique ( g ) and H λ We now prove although testing membership for C comprehensive = C GS ∩ C SA is co-NP complete,the community rule given by ( C Clique ( g ) ∩ C GS ∩ C SA ) can be constructive if g is small. Proposition 17.
Given a preference network A = ( V, Π) and a subset S ⊆ V , then we candetermine in O (2 g | S | g +3 ) time whether or not S ∈ ( C Clique ( g ) ∩ C GS ∩ C SA )( A ) . Particularly, if g = Θ(1) , then this decision problem is in P. However, the decision problem is co-NP completefor g = | S | δ for any constant δ ∈ (0 , .Proof. It takes time O ( | S | ) to check whether S ∈ C Clique ( g ) .Next we show that it takes time O ( | S | g ) to check if S ∈ C SA ( A ). Indeed, suppose G ′ ⊆ V − S is a witness that S / ∈ C SA ( A ). We claim that this implies that G ′ ⊂ π − s ([1 : | S | + g ]) ∀ s ∈ S .Suppose this is not true for some s ∈ S . Then ∃ v ∈ G ′ such that π s ( v ) > | S | + g , which in turnimplies that v ≺ π s u ∀ u ∈ G as π s ( u ) ∈ [1 : | S | + g ] ∀ u ∈ G . Thus there exists no bijection f s : S → G ′ with the property f − s ( v ) ≺ π s v , contradicting the assumption that G ′ ⊆ V − S isa witness that S
6∈ C GS ( A ). We can thus identify the set of all witnesses as follows: (1) Choose s ∈ S , and let T s = π − s ([1 : | S | + g ]) − S . (2) Choose a subset G ′ ⊆ T s . (3) Test if G ′ is a witnessthat S
6∈ C SA ( A ). First note that we are dealing with at most | S | g subsets. By Proposition 1,we can conduct the test of Step 3 performing | S | integer sorting. Thus, the total complexity forSteps 1-3 is O ( | S | g ).We can similarly test for group stability for S ∈ C Clique ( g ) . Suppose ( G, G ′ ) is a witness that S
6∈ C GS ( A ). Then, it must be the case that G ′ ⊂ π − s ([1 : | S | + g ]) ∀ s ∈ S − G . Supposethis is not true for some s ∈ S − G . Then there must be a v ∈ G ′ such that v ≺ π s u, ∀ u ∈ G as u ∈ π − s ([1 : | S | + g ]), which implies that there exists no bijection f s : G → G ′ with the28roperty f − ( v ) ≺ π s v , contradicting the assumption that ( G ⊂ S, G ′ ⊆ V − S ) is a witness that S
6∈ C GS ( A ).We say G ′ is a potential witness to S
6∈ C GS ( A ) if there exists G ⊂ S , such that ( G, G ′ ) is awitness to S
6∈ C GS ( A ). We can identify the set of all potential witnesses as follows: (1) Choose s ∈ S , and let T s = π − s ([1 : | S | + g ]) − S . (2) Choose a subset of G ′ ⊆ T s . (3) Test if G ′ isa potential witness. Again, we are dealing with at most | S | g subsets. As there are at most | S | | G ′ | ≤ | S | g candidates G to test for (using Proposition 1), Steps 1-3 takes at most O (2 g | S | g +3 )time.To show that for large g the problem of determining whether or not a set lies in C Clique ( g ) ∩C GS ∩ C SA is in co-NP, we reduce the problem to the one of determining whether for a givenpreference network A = ( V, Π), a set S ⊂ V is a member of ( C GS ∩ C SA )( A ). To define thereduction, we enlarge both V and S by a large, disjoint set ˜ S : V = V ′ ∪ ˜ S , S ′ = S ∪ ˜ S ,where ˜ S is chosen large enough to guarantee that g ( | ˜ S | ) ≥ | V | , implying in particular that | S ′ | + g ( | S ′ | ) ≥ | S ′ | + | V | ≥ | ˜ S | + | V | = | V ′ | . Due to this fact, we have that S ′ ∈ C Clique ( g ) ( V ′ , Π ′ )for all preference profiles Π ′ on V ′ . The statement now follows with the help of Lemma 2 andTheorem 8.Our final proposition in this subsection concerns the complexity of determining whether aset S lies in the class H λ . Proposition 18.
Given a preference network A = ( V, Π) and a subset S ⊆ V , we can determinein polynomial time whether S ∈ ( H λ ∩ C GS ∩ C SA )( A ) if (1 − λ ) | S | < , while it is co-NP completeto answer this question if (1 − λ ) | S | ≥ .Proof. We start with the proof of the positive statement. To this end, we first note that it takes( | V | − | S | ) | S | = O ( | V | ) comparisions to check whether S ∈ H λ .Next we show that if S ∈ H λ , then the only groups G ⊂ S that can threaten the stability of S are those for which | S − G | ≤ ⌊ (1 − λ ) | S |⌋ − . Indeed, assume that (
G, G ′ ) is a witness for S / ∈ C GS (Π), and let g = | G | . The assumption that S ∈ H λ then implies that for all ( u, v ) ∈ G × G ′ ⊂ S × ( V − S ), there are at most m = | S | − ⌈ λ | S |⌉ = ⌊ (1 − λ ) | S |⌋ elements s ∈ S − G ⊂ S such that v ≻ π s u . Thus the sum over all triples ( u, v, s ) ∈ G × G ′ × ( S − G )obeying this condition can be at most g m . On the other hand, if s lexicographically prefers G ′ over G , the number of pairs ( u, v ) ∈ G × G ′ ) obeying the above condition is at least g ( g +1)2 , givena lower bound of | S − G | g ( g +1)2 on the above number of triples. This proves that | S − G | ≤ gg +1 m, and hence | S − G | ≤ m −
1, where in the last step we used that both | S − G | and m are integers.Thus for (1 − λ ) | S | <
2, we may assume that G − S has size 1 (the case G − S = ∅ is trivial),which shows that there are at most | S | possible choices for G . Given G , we then only have tocheck whether a potential G ′ ⊂ V − S is lexicographically preferred to G by a single linear order π s , where s is the single element of S − G . Using Proposition 1, the existence of such a G ′ canbe checked by greedily choosing the first | G | elements of V − S with respect to π s . If this set islexicographically preferred to G , we know that S / ∈ C GS , and if for all G considered in the first29tep, the greedily found G ′ is not lexicographically preferred to G , S ∈ C GS . Since all S ∈ H λ are self-approving when λ > /
2, this completes the proof of the positive statement.To prove the negative statement, we use that it is NP-complete to determine whether in aformula consisting of 3-clauses, every clause is satisfied by exactly one literal in the clause, andthat this problem stays NP complete if we restrict ourselves to the case where each variableappears in exactly 3 clauses (cubic 1-in-3 SAT) [12]. Note that this means that we can partitionthe set of clauses into k = 7 classes such that the clauses in each class don’t share any variables(to see this, consider the graph obtained by joining two clauses whenever they share a variable;this graph has maximal degree at most 6, and hence can be colored by 7 colors, given the desiredpartition).Thus consider n boolean variables { x , . . . , x n } and k sets of 3-in-1 clauses C i such thatthe clauses in each C i have no common variables. We define X as the set of literals, X = { x , ¯ x , . . . , x n , ¯ x n } , and choose two additional sets Y and T , of size n and 2 k + 2, respectively.It will be convenient to label the elements of Y as y , . . . , y n , and the elements of T as 1 , . . . , k +2.Set S = Y ∪ T and V = S ∪ X, and choose Π ′ ∈ L ( V ) V of the form • If s ∈ Y , π ′ s ranks all of T first, followed by everyone in Y , followed by everyone in X • If s ∈ T , π ′ s ranks all of T first, then ranks V − T according to a yet to be determined π s ∈ L ( Y ∪ X ). • If v ∈ V − S , π ′ v is arbitrary.With this ranking, everyone in Y ranks all of S above all of V , showing that S ∈ H λ (( V, Π ′ ))as long as | S − Y | ≤ (1 − γ ) | S | , i.e., as long as (1 − γ ) | S | ≥ k + 1) = c . It also shows that S is self-approving, since everyone in S prefers all of T to all of V − S , which does not allow fora subset G ′ ⊂ V − S such that G ′ is lexicographically preferred to S by everyone in S . By thesame reasoning we also see that S is group-stable against any subgroup G which has a non-zerointersection with T . Finally, S is also stable against any subgroup G such that Y \ G = ∅ , sincefor such a subset, S − G contains an element s ∈ Y which prefers everyone in S to everyoneoutside S .Thus the only subgroup G against which S could be unstable is the set G = Y , i.e., S / ∈ ( H λ ∩C GS ∩C SA )( V, Π ′ ) if and only if there exists a subset G ′ ⊂ X such that G ′ is lexicographicallypreferred to G = Y by everyone in T . We now show that by defining Π appropriately, such a G ′ exists if and only if the the 1-in-3 SAT problem given by C , . . . , C k has a satisfying assignment.We first define π and π : π = [ x , ¯ x , y , . . . , x n , ¯ x n , y n ] π = [ x n , ¯ x n , y n , . . . , x , ¯ x , y ] . Clearly, G ′ is lexicographically preferred to G by both π and π if G ′ contains exactly one of x i and ¯ x i for each i . On the other hand, if G ′ is lexicographically preferred to G by π , then byProposition 1, G ′ must contain at least one of x and ¯ x , and if it is lexicographically preferredto G by π , it can contain at most one of x and ¯ x . Continuing by induction, we see that G ′
30s lexicographically preferred to G by both π and π if and only if G ′ contains exactly one of x i , ¯ x i for all i , i.e., if G ′ corresponds to a truth assignment for the variables x , . . . , x n .In a similar way, if C i consists of the clauses { z , z , z } , { z , z , z } , . . . , { z ℓ − , z ℓ − , z ℓ } ⊂ X , we define π i +1 = [ z , z , z , y , z , z , z , y , . . . , z ℓ − , z ℓ , z ℓ , y ℓ , Q ] π i +2 = [ Q, z ℓ − , z ℓ , z ℓ , y ℓ , . . . , z , z , z , y , z , z , z , y ] , where Q ranks everyone in X − { z , . . . , z ℓ } before the remaining elements y ℓ +1 , . . . , y n ∈ Y .Now the first ranking enforces that at least one literal of the clause { z , z , z } is chosen, whilethe last enforces that there is at most one such literal. Combining these two and continuing byinduction, we see that G ′ is lexicographically preferred to G by both π i +1 and π i +2 if and onlyif exactly one literal of each clause in C i is chosen.Putting everything together, we see that the 3-in-1 SAT problem has a satisfying assignmentif and only if S / ∈ ( H λ ∩ C GS ∩ C SA )( V, Π ′ ). Proposition 19.
Assume that n ≥ . There exists a preference network A = ( V, Π) such that C comprehensive ( A ) ≥ n/ .Proof. The preference profile, Π H & S , that is about to be described has been dubbed the “heroand sidekick” example as will soon become clear. Consider a world composed of n/ S , that is composed of all heroes and an arbitraryset of sidekicks. Note that because there are 2 n/ different sets of sidekicks, it is sufficient toshow that S is a community in C comprehensive ([ n ] , Π H & S ).First, note that S clearly satisfies SA .To show that S satisfies GS , consider two sets G ⊂ S and G ′ ⊂ V − S of equal size. We firstnote that it will be enough to consider the case where ( S − G ) × G contains no hero-sidekickpair ( u, v ), since otherwise u would prefer v over everyone else, in particular over everyone in G ′ .Applying this to the sidekicks in G , we conclude that G must contain at least as many heros assidekicks. On the other hand, G ′ can’t be lexicographically preferred to G if G contains at leasttwo heros, showing that only two cases are possible: G consisting of a hero-sidekick pair, or G made up of just a single hero. But neither one leads to a counter example if | S − G | > | G | = | G ′ | ,since then we can find an s ∈ S − G which is not the partner of any sidekick in G ′ , which meansthat s prefers the hero in G to everyone in G ′ . Since S contains all heros by assumption, we seethat S is group stable as soon as n ≥ In this section, we consider several stability measures and their impact on community structures.In particular, we focus on preference perturbations in Section 6.1 and the concept of stable fixedpoints of an aggregation function in Section 6.2. In both subsections, we will use B CT self-determined communities as our main examples to illustrate these measures. In Section 6.3, wewill study the structure of stable harmonious communities.31 .1 Community Stability with Respect to Preference Perturbations
We first study the structure of self-determined communities that remain self-determined evenafter a certain degree of changes in their members’ preferences.
Definition 10 (Preference Perturbations) . Let ∅ 6 = S ⊆ V , and let Π , Π ′ be two preferenceprofiles over V . For ≤ δ ≤ , we say Π ′ is a δ - perturbation of Π with respect to S if max v ∈ V (cid:12)(cid:12)(cid:8) i ∈ S : π i ( v ) = π ′ i ( v ) (cid:9)(cid:12)(cid:12) ≤ δ | S | . Given any community rule C and a preference network A = ( V, Π) , we say that a community S ∈ C ( A ) is stable under δ -perturbations if S ∈ C (( V, Π ′ )) for all Π ′ that are δ -perturbation of Π with respect to S . In other words, a preference profile is a δ -perturbation of another profile if, for each v ∈ V ,at most a δ -fraction of the members of S changed their preference of v . Recalling Definition 6,we now state our first stability result for B CT self-determined communities.
Proposition 20.
For any preference network A = ( V, Π) , if S ⊂ V is a B CT self-determinedcommunity that is stable under δ -perturbations, then ∃ α > δ such that S is an ( α, α − δ ) -B CTcommunity. Conversely, if S ⊂ V is an ( α, β ) -B CT self-determined community, then it isstable under ( α − β ) / -perturbations.Proof. Let u ∗ = argmin { φ Π S ( u ) : u ∈ S } , and let α = φ Π S ( u ∗ ) / | S | . We now prove that thecondition of the proposition implies α > δ . Suppose this is not true. Letting T = { s ∈ S : π s ( u ∗ ) ≤ | S |} , we have | T | = α | S | ≤ δ | S | . Now consider a preference profile Π ′ such that for s ∈ T , π ′ v shifts the ranking of u ∗ to n while maintaining the relatively rankings of all otherelements in π s , and π ′ v = π v ∀ v T . Then, as the ranking of u ∗ is more than | S | in every π ′ s for s ∈ S , we conclude that S is not a B CT self-determined community in ( V, Π ′ ), contradicting theassumption that S is stable under δ -perturbations. Now let v ∗ = argmax { φ Π S ( v ) : v ∈ V − S } ,and let β = φ Π S ( v ∗ ) / | S | . We can similarly show that if S is stable under δ -perturbations, then β < α − δ .The second direction of the proposition is straightforward.Thus, the main result of [3] can be restated as: there are at most n O (1 /δ ) B CT communitiesthat are stable under δ -perturbations. We further refine the stability studies of communityfunctions by introducing the notion of membership-preserving perturbation: Definition 11.
Let ( V, Π) be a preference network, and let ∅ 6 = S ⊂ V . A preference profile Π ′ on V is a membership-preserving perturbation of Π with respect to S if ∀ s ∈ S , π s ( S ) = π ′ s ( S ) . Note that the preference profile Π ′ considered in both Axiom CRNM and
CRM are specialcases of membership-preserving perturbations; in Axiom
CRNM , Π S and Π ′ S agree on S (i.e., forall s, u ∈ S , π s ( u ) = π ′ s ( u ), implying in particular π s ( S ) = π ′ s ( S )), and in Axiom CRM , Π S andΠ ′ S agree on V − S (i.e., for all s ∈ S, v ∈ V − S , π s ( v ) = π ′ s ( v ), implying π s ( V − S ) = π ′ s ( V − S ))and hence also π s ( S ) = π ′ s ( S ). Theorem 11.
For any preference network A = ( V, Π) , the number of B CT communities thatare stable under membership-preserving, δ -perturbations of Π is polynomial in n O (1 /δ ) . roof. It will be sufficient to show that if a B CT community S is stable under membership-preserving, δ -perturbations of Π, then either1. S is an ( α, α − δ )-B CT community for some α > δ , or2. ∃ s ∈ S such that π s ( S ) = [1 : | S | ].Indeed, in the first case, there are at most n O (1 /δ ) many ( α, α − δ )-B CT communities by [3],and in the second case, we have that all communities are of the form S = π − v ([1 : k ]) for some s ∈ V and k ∈ [ n ], showing that there are at most n such communities.To establish the above statement, letting u ∗ = argmin { φ Π S ( u ) : u ∈ S } and α = φ Π S ( u ∗ ) / | S | ,we now prove that if S is not an ( α, α − δ )-B CT community, then there must be s ∈ S , π s [1 : | S | ] = S . The assumption that S is not an ( α, α − δ )-B CT community implies that φ Π S ( v ∗ ) / | S | ≥ α − δ where v ∗ = argmax { φ Π S ( v ) : v ∈ V − S } . Let T = { s ∈ S : π s ( v ∗ ) ≤ | S |} .Then, | T | = φ Π S ( v ∗ ) ≥ ( α − δ ) | S | . Since S is a B CT community, we know that | T | < α | S | .Using these conditions, we now define a perturbed preference profile. Key to our constructionis the following observation: For each s ∈ S − T , if π s ( S ) = [1 : | S | ], then ( V − S ) ∩ π s [1 : | S | ] = ∅ .Thus, there exists π ′ s that agrees with π s on S and π ′ s ( v ∗ ) ≤ | S | – we can simply swap v ∗ withany element in ( V − S ) ∩ π s [1 : | S | ]. Thus, either there exists s ∈ S − T such that π s ( S ) = [1 : | S | ](which implies Case 2 above), or π s ( S ) = [1 : | S | ] , ∀ s ∈ S − T . The latter implies that we canfind a set ˜ S ⊂ S − T of size α | S | − | T | ≤ δ | S | and a membership-preserving, δ -perturbations Π ′ ofΠ such that π ′ s ( v ∗ ) ≤ | S | for all s ∈ T ∪ ˜ S , implying that S is not a B CT community in ( V, Π ′ ).This contradicts the assumption that S is stable under membership-preserving, δ -perturbationsof Π. We can also strengthen the concept of fixed points in our social choice based community frame-work. Particularly, we measure the stability of a community defined by a fixed-point ruleaccording to some variation of the following definition.
Definition 12. ( δ -Strong Fixed Points) Let A = ( V, Π) be a preference netowrk, F : L ( V ) ∗ → L ( V ) be a preference aggregation function, and δ ∈ [0 : 1] be a coherence parame-ter. Then, S ∈ C F (( V, Π)) is δ -strong if for ∀ T ⊆ S such that | T | ≥ (1 − δ ) · | S | , u ≻ F (Π T ) v ∀ u ∈ S, v ∈ V − S. Our goal is to understand the influence of a preference aggregation function F and thestability parameter δ (0 ≤ δ ≤
1) on the structure of the δ -strong F -self-determined communities.Before discussing this further, we point out some subtleties that arise when applying Defini-tion 12 to general aggregation functions. We illustrate this subtlety using weighted fixed-pointrules, and, in particular, by comparing the community rule defined by the B CT voting functionto that defined by the Borda count.Recall that in Definition 8, for preference networks with n elements, a preference aggregationfunction is determined by a sequence of weighting vectors W = ( w , w , . . . ) where w k ∈ R n ,denotes the weights for the aggregation of k preferences. While this weight vector is independentof k for the Borda count, it in general can be different for each k , and indeed does depend33n k for B CT voting. Concretely, for the Borda count, every voting member gives scores n, n − , . . . , V , while in B CT voting, it gives a score of 1 to the first k inher preference list, making her scores dependent on the total number of voters, k . Thus whendefining self-determined communities with the Borda count, one does not need to first anticipatethe community size before aggregating the preferences of its members, but when defining self-determined communities with B CT voting, the weight assigned to an element by a preferencedepends on the size of the subset under consideration.In this regard, when measuring the stability of a community S , Definition 12 uses the sameweighting vector to evaluate F (Π T ) and F (Π S ) for the Borda count based community rule,while it uses different weighting vectors to evaluate F (Π T ) and F (Π S ) for the B CT communityrule, and these weighting vectors depend on | T | . Thus, the former application of Definition 12appears more natural than the latter application.As a result, we will use the following variation of Definition 12 to measure the strength of aB CT community.
Definition 13 ( δ -Strong B CT Communities) . For each T ⊆ V and i ∈ V , let φ Π T,k ( i ) denotethe number of votes that member i would receive if each member s ∈ T were casting a vote foreach of its k most preferred members according to its preference π s .For δ ∈ [0 : 1] , a non-empty set S ⊆ V is a δ -strong B CT community in A = ( V, Π) if ∀ u ∈ S, v ∈ V − S and T ⊆ S such that | T | ≥ (1 − δ ) · | S | , φ Π T, | S | ( u ) > φ Π T, | S | ( v ) ∀ u ∈ S, v ∈ V − S. Proposition 21. If S is a δ -strong B CT community of A = ( V, Π) , then ∃ α ≥ δ such that S is an ( α, α − δ ) -B CT community.Proof.
Let u ∗ = argmin { φ Π S ( u ) : u ∈ S } , and let α = φ Π S ( u ∗ ) / | S | and let v ∗ = argmax { φ Π S ( v ) : v ∈ V − S } , and let β = φ Π S ( v ∗ ) / | S | . We now prove that α − β > δ .The pair u ∗ and v ∗ partitions S into four subsets. S = { s ∈ S : ( π s ( u ∗ ) [1 : | S | ]) and ( π s ( v ∗ ) [1 : | S | ]) } S = { s ∈ S : ( π s ( u ∗ ) [1 : | S | ]) and ( π s ( v ∗ ) ∈ [1 : | S | ]) } S = { s ∈ S : ( π s ( u ∗ ) ∈ [1 : | S | ]) and ( π s ( v ∗ ) [1 : | S | ]) } S = { s ∈ S : ( π s ( u ∗ ) ∈ [1 : | S | ]) and ( π s ( v ∗ ) ∈ [1 : | S | ]) } Then | S | + | S | + | S | + | S | = | S | , | S | + | S | = β · | S | , | S | + | S | = α · | S | , | S | + 2 | S | + | S | < (1 − δ ) · | S | , where the last inequality follows from the assumption that S is a δ -strong B CT-self-determinedcommunity.To see this, we first note that | S | < | S | due to the fact that β < α . Define T to be the unionof S , S , S , and ˜ S , where ˜ S ⊂ S is an arbitrary subset of size | S | . Assume by contradiction34hat | T | ≥ (1 − δ ) | S | . Since S is a δ -strong B CT-self-determined community, this would implythat φ T, | S | ( u ∗ ) > φ T, | S | ( v ∗ ), i.e.0 < X s ∈ T (cid:16) π s ( u ∗ ) ≤| S | − π s ( v ∗ ) ≤| S | (cid:17) . But the right hand side is equal to | ˜ S | − | S | = 0, leading to a contradiction. Therefore, | T | < (1 − δ ) · | S | , as claimed.Subtracting the fourth of the above equations from the first, we obtain ( | S | + | S | + | S | + | S | ) − ( | S | + 2 | S | + | S | ) = ( | S | + | S | ) − ( | S | + | S | ) > | S | − (1 − δ ) · | S | = δ · | S | . Thus, bythe second and third equation, we have α · | S | − β · | S | > δ · | S | . Proposition 22.
For any δ ∈ (0 , , the number of δ -strong B CT communities in any prefer-ence network is n O (1 /δ ) .Proof. This follows from the main result of [3] and Proposition 21 above.
Applying the stability notions of Sections 6.1 and 6.2, we define two types of stable harmoniouscommunities. Before doing so, we recall the definition of harmonious communities, Definition 5,and the definition of λ -harmonious communities from Rule 5. Definition 14 (Stable Harmonious Communities) . For δ ∈ [0 : 1 / , a non-empty subset S is a δ -stable harmonious community in A = ( V, Π) if S is ( δ + 1 / -harmonious, i.e., if ∀ u ∈ S, v ∈ V − S , at least (1 / δ ) -fraction of { π s : s ∈ S } prefer u over v . For δ ∈ [0 : 1] , S is a δ -strongharmonious community in A if ∀ u ∈ S, v ∈ V − S and T ⊆ S such that | T | ≥ (1 − δ ) · | S | , themajority of { π s : s ∈ T } prefer u over v . Note that a δ -stable harmonious community is not quite the same as a harmonious com-munity stable under δ -perturbations as defined in Section 6.1. Instead, we have that a δ -stableharmonious community is a harmonious community that is stable under any δ ′ -perturbations aslong as δ ′ < δ/
2, and that conversely, a harmonious community that is stable under δ pertur-bation is a δ -stable harmonious community. By contrast, the definition of δ -strong harmoniouscommunities maps exactly to the definition given in Section 6.2. Proposition 23. If S is a δ -strong harmonous community, then S is a δ/ -stable harmoniouscommunity.Proof. For each pair u ∈ S, v ∈ V − S , let f ( u, v ) = |{ s ∈ S : u ≻ π s v }| − |{ s ∈ S : u ≺ π s v }| be the preference gap between u and v with respect to S . Suppose ( u ∗ , v ∗ ) = argmin { f ( u, v ) : u ∈ S, v ∈ V − S } . We now show that if S is a δ -strong harmonous community of A , then f ( u ∗ , v ∗ ) > δ · | S | . The pair u ∗ and v ∗ partitions S into two subsets. S ≻ = { s ∈ S : u ∗ ≻ π s v ∗ } and S ≺ = { s ∈ S : u ∗ ≺ π s v ∗ } . We have | S ≻ | + | S ≺ | = | S | and | S ≻ | > | S ≺ | . Let T be the unionof S ≺ and | S ≺ | arbitrary members of S ≻ . Since members of T are indifferent about u ∗ and v ∗ ,we have | T | = 2 | S ≺ | ≤ (1 − δ ) · | S | . Thus f ( u ∗ , v ∗ ) = | S ≻ | − | S ≺ | = ( | S ≻ | + | S ≺ | ) − | S ≺ | ≥| S | − (1 − δ ) · | S | = δ · | S | . Thus, | S ≻ | > (1 / δ/ · | S | , and at least (1 / δ/ S prefer u ∗ over v ∗ . 35ith a simple probabilistic argument, we can bound the number of δ -stable harmoniouscommunities in any preference networks. Theorem 12. ∀ δ ≤ / , the number of δ -stable harmonious communities in any preferencenetwork is n
12 log n/δ .Proof. Let S be a δ -stable harmonious communities. For any multi-set T ⊆ S , we say T identifies S if for all u ∈ S and v ∈ V − S , the majority of T prefer u to v . Note that such a T determines S once the size of S is set. To see this, note that the condition implies that u ≻ F (Π T ) v for all( u, v ) ∈ S × ( V − S ), which in turn implies that S is of the form V ∪ · · · ∪ V i where ( V , V , . . . )are the components of the ordered partition F (Π T ), ordered in such a way that V ≻ F (Π T ) V , ... (see Proposition 10 and its proof). Thus once F (Π T ) and the size of S are fixed, S is uniquelydetermined.We now show that ∃ T ⊂ V of size 12 log n/δ that identifies S . To this end, we consider asample T ⊂ S of k = 12 log n/δ randomly chosen elements (with replacements). We analyzethe probability that T identifies S . Let T = { t , ..., t k } , and for each u ∈ S and v ∈ V − S , let x ( u,v ) i = [ u ≻ π ti v ], where [ B ] denotes the indicator varable of an event B . Then T identifies S iff P ki =1 x ( u,v ) i > k/ , ∀ u ∈ S, v ∈ V − S . We now focus on a particular ( u, v ) pair and boundPr hP ki =1 x ( u,v ) i ≤ k/ i . We first note that E " k X i =1 x ( u,v ) i = k X i =1 E h x ( u,v ) i i ≥ (cid:18)
12 + δ (cid:19) · k. By a standard use of the Chernoff-Hoeffding boundPr " k X i =1 x ( u,v ) i ≤ k/ ≤ Pr " k X i =1 x ( u,v ) i ≤ (1 + 2 δ ) − E " k X i =1 x ( u,v ) i ≤ Pr " k X i =1 x ( u,v ) i ≤ (1 − δ ) E " k X i =1 x ( u,v ) i ≤ e − δ (1 / δ ) k ≤ e − δ k ≤ n , where we used that (1 + 2 δ ) − = 1 − δ (1 + 2 δ ) − ≤ − δ in the third step.If T fails to identify S , then there exists ( u ∈ S, v ∈ V − S ) such that P ki =1 x ( u,v ) i ≤ k/
2. Asthere are at most | S || V − S | ≤ n such ( u, v ) pairs to consider, by the union bound,Pr [ T identifies S ] ≥ − X u ∈ S,v ∈ V − S Pr " k X i =1 x ( u,v ) i ≤ k/ > − /n> . Thus, if S is a δ -stable harmonious communities, then there exists a multi-set T ⊂ V of size12 log n/δ that identifies S . We can thus enumerate all δ -stable harmonious communities by36numerating all ( T, t ) pairs, where T ranges from all multi-subsets of V of size 12 log n/δ and t ∈ [1 : n ] and check if T can identify a set of size t . While the results of this paper are conceptual and are built on the abstract framework ofpreference networks, we hope this study is a significant step towards developing a rigorous theoryof community formation in social and information networks. In particular, we hope this willbe used to inform and choose among other approaches to community identification which havebeen developed. Below we discuss a few short-term research directions that may help to expandour understanding in order to make more effective connection with community identification innetworks that arise in practice.
Preferences Models
We have based our community formation theory on the ordinal concept of utilities used in socialchoice and modern economic theory [2]. The resulting preference network framework, like thatin the classic studies of voting [2] and stable marriage [7], enables our axiomatic approach tofocus on the conceptual question of network communities rather than the more practical questionof community formation in an observed social network. To better connect with the real-worldcommunity identification problem, we need to loosen both the assumption of strict ranking andthe assumption of complete preference information.With simple modifications to our axioms, we can extend our entire theory to a preferencenetwork A = ( V, Π) that allows indifferences , i.e., Π is given by n ordered partitions { π , ..., π n } : π i ∈ L ( V ). This extension enables us to partially expand our results to affinity networks. Recallan affinity network A = ( V, W ) is given by n vectors W = { w , ..., w n } , where w i is an n -placenon-negative vectors. We can extract an ordinal preference π i ∈ L ( V ) from the cardinal affinitiesby sorting entries in w i – elements with the same weight are assigned to the same partition.Although this conversion may lose some valuable affinity information encoded in the numer-ical values, it offers a path for us to apply our community theory – even in its current form –to network analysis. For example, as suggested in [3], given a social network G = ( V, E ), wecan first define an affinity network A = ( V, W ) where w i is the personalized PageRank vector ofvertex i , and then obtain an preference network ( V, Π) where π ∈ L ( V ) ranks vertices in V by i ’s PageRank contributions [1] to them.Theoretically, we would like to extend our work to preference networks with partially or-dered preferences as a concrete step to understand community formation in networks with in-complete or incomparable preferences. Like our current study, we believe that the existingliterature in social choice – e.g., [16] – will be valuable to our understanding. We expect thatan axiomatic community approach to preference networks with partially ordered preferences,together with an axiomatization theory of personalized ranking in a network, may offer us newunderstanding of how to address the two basic mathematical problems – extension of individ-ual affinities/preferences to community coherence and inference of missing links – for studyingcommunities in a social and information network. As this part of community theory becomessufficiently well developed, well-designed experiments with real-world social networks will benecessary to further enhance this theoretical framework.37 tructures, Algorithms, and Complexity Our taxonomy theorem provides the basic structure of communities in a preference network,while the coNP -Completeness result illustrates the algorithmic challenges for community iden-tification in addition to community enumeration. On the other hand, our analysis of the har-monious rule and the work of [3] seem to suggest some efficient notion of communities can bedefined.However, it remains an open question if there exists a natural and constructive communityrule that simultaneously (i) satisfies all axioms, (ii) allows overlapping communities, and (iii)has stable communities which are polynomial-time samplable and enumerable.
Acknowledgements
We thank Nina Balcan, Mark Braverman, and Madhu Sudan for all the insightful discussions.The complexity proof of Proposition 18 is due to Madhu.
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