Characterizing Attitudinal Network Graphs through Frustration Cloud
CCharacterizing Attitudinal Network Graphs throughFrustration Cloud
Lucas Rusnak Jelena Teˇsi´cJuly 27 2020
Abstract
Attitudinal Network Graphs (ANG) are network graphs where edges capture anexpressed opinion: two vertices connected by an edge can be agreeable (positive) orantagonistic (negative). Measure of consensus in attitudinal graph reflects how easyor difficult consensus can be reached that is acceptable by everyone. Frustration indexis one such measure as it determines the distance of a network from a state of totalstructural balance. In this paper, we propose to measure the consensus in the graph byexpanding the notion of frustration index to a frustration cloud , a collection of near-est balanced states for a given network. The frustration cloud resolves the consensusproblem with minimal sentiment disruption , taking all possible consensus views overthe entire network into consideration. A frustration cloud based approach removesthe brittleness of traditional network graph analysis, as it allows one to examine theconsensus on entire graph. A spanning-tree-based balancing algorithm captures thevariations of balanced states and global consensus of the network, and enables us tomeasure vertex influence on consensus and strength of its expressed attitudes. The pro-posed algorithm provides a parsimonious account of the differences between strong andweak statuses and influences of a vertex in a large network, as demonstrated on sampleattitudinal network graphs constructed from social and survey data. We show thatthe proposed method accurately models the alliance network, provides discriminantfeatures for community discovery, successfully predicts administrator election outcomeconsistent with real election outcomes, and provides deeper analytic insights into ANGoutcome analysis by pinpointing influential vertices and anomalous decisions.
In Attitudinal Network Graphs (ANG) edge weights capture an attitude of one vertex to-wards the other vertex. Attitude is expressed as a sentiment of (dis)approval of anothervertex or its action as edge weight and/or edge sign. Social ANG examples are team mem-bers evaluation in the company, student evaluations of instructor, movie recommendationbased on common interests, or a “trustworthiness” of a product reviewer or seller in onlinestores. These sentiments have tangible real-world effects, such as annual performance scoresand promotions in the corporation. If there is a group decision to be made, consensus (ma-jority voting) on the sentiment or some other simplified rule guides the decisions of the final1 a r X i v : . [ c s . S I] S e p utcome in such networks. Graph decision algorithms are rarely scrutinized, as consensusand majority voting is established social construct [30]. Only when the individual outcomesare known an individual’s status may be perceived as elevated or diminished relative to theirpeers, and only anecdotally questioned or explained. Sensitivity to bias, fraud, and false-hood of such algorithms has been put under magnifying glass in the last couple of years,as state-of-art research examined controversy of the decisions points in cases of status quo[31] or subgroup mobilization against other groups [27]. Consensus in Attitudinal NetworkGraphs (ANG) translates to reaching structural balance of underlying signed graph. Struc-tural balance introduces the bipartite structure, and recent research has proposed Laplaciandynamics to show convergence of the graphs to any balanced state [4, 44]. In this paper,we discover all different balanced states ANG can have, and characterize consensus as theability of vertices in underlying signed graph to reach any of the discovered balanced states.Attitudinal Network Graphs (ANG) broadens the definition of a social network graphto connections (edges) among people, content, AI modules, and groups (vertices) with theattitudes captured by edges between two vertices as agreeable (positive edge), non-existent(no edge), or antagonistic (negative edge). Neutral edges are ambivalent edges, and inmathematical sociology are equivalent to non-existent relations (no edge present) or opposingpositive and negative edges between vertex [1]. In this work, we consider only agreeable andantagonistic edges between two vertices, and model ANG as undirected signed graph.A signed graph consists of a collection of vertices that are linked together with edges, andeach edge receives a sentiment value that expresses the opinion shared between the vertices.We model the agreement and positive sentiment between 2 vertices as “+1”, and disagree-ment or general negative sentiment between 2 vertices as “-1”. Note that we do expand ourmodeling to ambivalent and directed edges in the consequent work. Fritz Heider introducedBalance Theory for signed graphs in 1948 [21]. Balance theory examined consensus in triadicrelationships in a signed triangle graph. Figure 1 shows all possible edge signs for a signedtriangle graph: these eight graphs are different as they have different edge signs betweenspecific vertices (top, left, and right). Each of eight graph signing is a state of a trianglegraph. Balance Theory is the base model of attitude change analysis among three personsin signed graph [21]: a triangle state is considered balanced if the product of the edge signsis positive and unbalanced if the product of edge signs is negative. A balanced triadic rela-tionship in a signed graph is captured as “the enemy of my enemy is my friend” paradigmin mathematical social modeling. Figure 1: Sign graph triangle: top row are balanced states, and bottom row are unbalanced states.
Triangle signed graph can have four balanced states (top row of Figure 1) and fourunbalanced states (bottom row of Figure 1). There are four different ways to achieve a2alanced state in the triangle, and emphasis here is on the fact that there is more thanone balanced state.
There can be multiple consensus scenarios for a single graph, and theycan all capture different aspects of reaching consensus . An individual balanced state is asnapshot of balanced assessment of the network, and it is not sufficient as all sentiments arenot necessarily equal, as illustrated in Figure 1 top row. In this paper, we implement analgorithm that considers the collective of nearest consensus states to characterizes networkgraph behavior. This type of analysis considers all possible consensus outcomes of attitudinalnetwork graph for a complete consensus characterization.
Mathematical sociology was introduced as a field when Rashevsky characterized large socialnetworks as graphs, where vertices are persons and edges measure the level of acquaintance-ship [38]. Attitudinal graph networks capture the attitude expressed as an edge sign in thenetwork graph. The mathematical foundation of signed graphs [18] and social balance the-ory [1, 21] introduced the concepts of modeling balance and agreement in social networksusing more complex mathematical models. The attitudinal model based on a rudimentaryalgebraic framework was introduced by Abelson and Rosenberg [1]. Harary introduced frus-tration index of a signed graph as a measure of how far the network graph is from a state ofstructural balance [19]. Harary’s proposed measure is the smallest number of edges whosenegation in network graph results in a balanced signed graph. Harary’s attitudinal balancedmodel was formalized in graph-theoretic terms [9], and fully characterized by Zaslavsky [52] inmatroid-theoretic terms. Davis [11] studied necessary and sufficient conditions for clusteringof attitudinal graph. Mathematical sociology modeling has evolved to address sociologicalphenomena in various fields of social science (culture, social psychology) and to help un-derstand, evaluate, and predict patterns of social relationships and interactions [23]. Largevirtual communities and decision networks of the 21 st century initiated the explosive growthof social network analysis and network science fields. The analysis of largely digital traces of social networks at scale expanded well-studied mathematical algorithms for reinforcement,information processing, social judgment, balance, and dissonance [23]. Wasserman et al.introduced the social network analysis as algebraic graph representations and proposed aseries of statistical tests [49]. The domain research has focused on predicting the existenceand/or sentiments of edges in the graph, recommending content or a product, or identifyingunusual trends. Baseline signed graph theory was used to explain relative status individ-uals hold within in a social network [28, 29] and focused on socially-conscious science tohelp understand bias, controversy, conflict, and trust [16, 34]. All mathematical models inthe network science that model intents and trends in online social networks have relied onaspects of well established consensus-based models in signed graph theory [10, 14, 51] andbalanced modeling [24, 33, 40, 46, 53, 54].Scientists have proposed a multitude of measures to access the rich information coded inthe network graph. The digital traces of attitudinal network graphs (ANG) are noisy. Everyvote, review, and evaluation does not necessarily reflect true sentiment: it can be a random,fraudulent, wrong, or mood driven (if it is a person). Bias is a favoritism, nepotism, orsome kind of unfair advantage. Mishra et al. [34] introduces trustworthiness and deserve aslocal vertex-based measures of bias and reflect the expected weights of out- and in- edges.3 ontroversy was introduced by Garimella et al. [14] as the likelihood a random walk willreturn to the same side of the network. This method improves the examination of triangles in[28] by including pendant vertices, and propose to reduce controversy by bridging opposingviewpoints.
Conflict is defined in Chen et al. [10] by examining the Laplacian matrix toproduce a “Conservation Law of Conflict” reminiscent of Kirchhoff’s laws. Kumar et al.[27] discuss group mobilization against other group to describe conflict in intra-communityinteractions, and Guha et al. in [16] examines trust through iterative build of belief matrix.Yuan et al. [51] introduces a sign prediction model for sparse data edge prediction wherethey convert the original graph into the edge-dual graph and apply machine learning topredict signs in sparse graphs. Controllability and consensus algorithms on networks stemsfrom Altafini [4] by examining the effects of bipartite consensus of Harary [18]. Pan et al.[37] examined the bipartite structure of via Laplaican dynamics and mode decomposition.Hu et al. shows that the ideal state of the multi-agent system can modeled as a balancedgraph, and that the system converges to the optimal state through the bipartite consensusiterations [22], while uncontrollability and stabilizability was examined by Alemzadeh etal. in [3]. Algorithms for the characterization of status quo has been examined in [31]for transitive graphs. Jiang et al. proposes a sign-driven consensus as a control protocolmeasured via Laplacian dynamics [25]. She et al. [44] examines consensus in terms ofgraphical characterizations of the controllability of signed networks, and offers a heuristicalgorithm for leader selection based on balance theory. Network models of late conceptualizeattitudes as networks consisting of evaluative reactions and interactions intra-network ascollective dynamic reactions are often too complex for existing tools to analyze the entirenetwork. Established methods of network graph analysis focus on endorsement analysisthrough local topology analytics and strive for agreement by changing [28], adding [14] orremoving [16] edges in the graph.What is common for the related research is that it interprets ”balance state” as oneitem, and that there is no consensus (pun intended) among researchers on how to accessthe robustness, resilience, and reliability of the network algorithm outcomes, or how toidentify anomalies and a network’s controversy makeup. There is a clear need to measureperformance of the algorithms that define outcomes, characterize consensus in social or multi-agent attitudinal networks as a unit, and access vertex and edge contribution to graph as awhole. We introduce a new discrete methodology that finds all the nearest consensus-drivenbalanced states of an attitudinal network and quantifies their relative importance to theinitial signed network.
We introduce an algorithm based on mathematical modeling of balanced social interaction,and demonstrate that the method can aid in identifying and quantifying influences and flagquestionable outcomes in objective and robust manner. Proposed graph specialization ofbalance theory expands on triadic and community analysis, and model vertices’ attitudinalstrength and influence on an entire network.In signed graph theory the “frustration index” is defined as the smallest number of edgeswhose change in sign can result in a balanced signed graph [19], and has been shown toplay a critical role in the investigation of ground states of the disordered system [2, 5]. We4eneralize the notion of the frustration index to the frustration cloud as we consider any minimal balancing set, and not just a smallest one. All minimal balancing sets form a multisetof nearest balanced states to a given signed graph called the frustration cloud . To quantifylevels of agreement in the network, we sample the frustration cloud via associated family ofbalanced matroidal bases characterized in [52] to correctly model statistical significance ofall minimal balancing sets in the frustration cloud. This statistically meaningful samplingof frustration cloud produces a robust way to handle the brittleness of the data space forANG data and avoid challenges presented in [43]. Our graphB (short for graph b alancing)approach combines the requirement for statistical parity across the nearest balanced stateswith the requirement to consider all vertices instead of few selected ones, and relies on thespanning trees, not random walks [14]. Next, the sentiments are reconstructed around aspanning tree to produce a set of nearest balanced states. The resulting balanced state isa generalization of bipartite graphs [6, 19], and the resulting negative edge cut defines twoconsensus-based sets.Graph characterization metrics are derived from weighed sampling of nearest balancedstate of a given graph: status quantifies an individuals contribution to reaching consen-sus over weighted frustration cloud, agreement quantifies the edge contribution to reachingnearest balanced state; influence quantifies the edge contribution to reaching consensus,and controversy provides ANG characterization. We demonstrate the implementation andsuperiority over spectral clustering of the proposed method using survey dataset [39]. Wedemonstrate the effectiveness of the approach in discovering contentious outcomes and dis-covering status and influence on Wikipedia administrator and Slashdot datasets [30].Laplacian dynamics and spectral techniques are often used to produce a single conver-gent balanced state [3, 4, 22, 25, 37, 44]. We propose a new discrete alternative to Laplaciandynamics to find all nearest balanced states, and to spectral clustering to identify the con-nected subgroups in the ANG. There is no consensus among researchers on how to accessthe robustness, resilience, and reliability of the network algorithm outcomes, or how to iden-tify anomalies and a network’s controversy makeup. We propose a methodology that forany ANG: (1) determines all the nearest balanced states via basis sampling via spanningtrees; (2) quantifies the importance of each balanced state relative to the likelihood it willbe become the consensus state; (3) quantifies an individuals status relative to their peers;(4) characterizes the potential maximum status of an individual over tie-break scenarios;(5) provides a constant metric of controversy for the entire network that is subject to aConservation Law; (6) quantifies an individual decision or opinion based on agreement ; (7)aggregates agreement to each individual to quantify influence over others; and (8) comparesstatus and influence to quantify the positive/negative relationship of the entire network andprovide a spectrum of status-influence. In the event that the sentiment data provided is re-lated to promotions, and the outcome of those promotions are known, the research examinesthe efficacy of this new methodology by outcomes as status separates “promoted” from “notpromoted” and identifies any outliers in either case, where the influence separates “voters”from those “voted on”. 5
Signed Graphs and Balance Theory A signed graph Σ is a pair ( G, σ ) that consists of a graph G = ( V, E ) and an edge-signingfunction σ ∶ E → {+ , − } . For a set of edges E in a signed graph Σ let E + (resp. E − )denote the set of positive (resp. negative) edges of G — the signs of the edges are regardedas sentiments between two vertices. A signed graph is balanced if the product of the signsin every circle is positive [9, 18]. A switching function on a signed graph is any function ς ∶ V → {− , + } , and switching a signed graph Σ = ( G, σ ) by ς is the signed graph Σ ς = ( G, σ ς ) where σ ς = ς ( v i ) − σ ( e ij ) ς ( v j ) . Signs graphs adhere to the following two lemmas and onecorollary [19, 52]: Lemma 2.1.
Switching is an equivalence relation on the set of signed graphs on an graph.
Lemma 2.2.
Switching does not alter the sign of any circle in the signed graph.
Corollary 2.2.1.
The set of balanced signed graphs on a given graph are switching equivalent. ⇒ ⇒
Figure 2: Switching the red vertex preserves the signs of cycles, as specified by Lemma 2.1, andCorollary 2.2.1 A balancing set of Σ is a set of edges whose sign reversal produces a balanced signedgraph. A balancing set is minimal if no proper subset is a balancing set. The frustrationindex of a signed graph Σ, denoted F r ( Σ ) , is the smallest number of edges whose deletion(equivalently, negation) results in a balanced signed graph, as defined by Harary [19]. We canswitch a signed graph to have the smallest number of negative edges possible to determine itsfrustration. As a corollary, the negative edges in this switched state represent the minimumnumber of edges whose sign reversals would produce a balanced signed graph in the original. Lemma 2.3.
For a signed graph Σ with ne ( Σ ) negative edges, let [ Σ ] mark its switchingclass. The frustration index of signed graph Σ is then: F r ( Σ ) = min Σ ′ ∈[ Σ ] ne ( Σ ′ ) . All balanced signed graphs have a frustration of 0 since they can be switched into theall-positive signed graph. Next, we present equivalent formulations of balance that will beused in the formulation our new consensus model.
Theorem 2.4 ([18, 19]) . For a signed graph Σ , the following are equivalent:1. Σ is balanced. (All circles are positive.) . For every vertex pair ( v i , v j ) , v i , v j ∈ V all ( v i , v j ) -paths have the same sign. (Agree-ment/Consensus.)3. There exists a bipartition of the vertex set into sets U and W such that an edge isnegative if, and only if, it has one vertex in U and one in W . The bipartition ( U , W )is termed Harary-cut .4.
F r ( Σ ) = . ( frustration.) Figure 3: Balanced signed graphs with their Harary-cut of negative edges represented by dashededges.
Figure 3 shows all possible balanced states on given underlying graph. Each of balancedstates satisfies all conditions on Theorem 2.4. The Harary-cuts (bipartition of graph in twosets) are emphasized by having the negative edges represented by dashed edges. A spanningtree T of an undirected graph G is a maximal acyclic subgraph that contains all the verticesof G . For a spanning tree T in graph G and an edge e ∈ E ( G ) ∖ E ( T ) , the fundamental cycleof e with respect to T in G is the unique cycle in T ∪ e . The more general signed graphiccharacterization from [52] is Theorem 2.5.1. Theorem 2.5.
The bases of the signed graphic frame matroid are spanning trees plus anadditional edge that makes a negative fundamental cycle.
All fundamental cycles are positive in balanced graphs, giving Corollary 2.5.1:
Corollary 2.5.1.
The bases of a balanced signed graph are spanning trees.
Corollary 2.5.1 reaffirms the connection between balanced states and the all-positivesigned graph. Sentiment analysis in attitudinal networks are trivialized by this corollary, asthe claim that a connected signed graph Σ is balanced if, and only if, is an eigenvalue of L Σ is a direct result of the switching nature of balanced graphs and their Laplacians. Theglobal consensus state in Attitudinal Graph Networks is switching equivalent to balancedstate in signed graph theory [52]. 7 .1 Signed Graph Tree Balancing Algorithm Mathematical sociologists have relied on balance theory [21] to model a suite of social inter-actions. Harary described how balance theory can predict coalition formation, see Theorem2.4: if a signed graph is balanced there will be a tendency for the status quo; if it is notbalanced, each of the edges in the cycle need to be examined relative to the strength of thecycle [18]. From theorem to algorithm, that translates to Algorithm 1. For a connectedgraph G , let Σ = ( G, σ ) be the signed graph of G , and T G be the set of spanning trees of G . Balancing algorithm constructs a function from the set of balanced matroidal bases ofΣ (the spanning trees of G ) to balanced signed graphs, as outlined in Algorithm 1. Theunderlying graph G is assumed to be connected. If it is not, the algorithm is applied toconnected components of G . Algorithm 1
Signed Graph Tree-Balancing Algorithm:
Input:
Input signed graph Σ = (
G, σ ) . Input:
Select T ∈ T , T is a spanning tree of Σ for all edges e , e ∈ Σ ∖ T doif fundamental cycle T ∪ e is negative then change edge sign: e − − > e + ; e + − > e − end ifend for Construct new signed graph Σ
Output:
Balanced Graph Σ T An example of Algorithm 1 is illustrated in Figure 4. The signed graph Figure 4(left)has 8 spanning trees visualized with darker edges in Figure 4(right). The edges outside eachspanning tree are indicated by dashed edges. For any spanning tree T and an edge e outsidethat tree, the sub-graph T ∪ e contains a unique fundamental cycle C . The sign of e is chosenso that C is positive . ⇒ Figure 4: The spanning trees of a signed graph (bold) produce a balanced signed graph (re-signededges orange and teal), with isomorphic balancing grouped (boxes).
Consider the original graph Figure 4(left) and top right balancing graph in this example.The spanning tree is ⊔ shaped, and those edges do not change sign. The sign of the diagonaledge is originally negative. In the fundamental cycle diagonal edge forms with the ⊔ shapedspanning tree the other two edges are positive. The edge needs to change sign to positiveto have the cycle positive. Next, the top edge was originally positive, and it forms a squarewith the ⊔ shaped spanning tree and for the cycle to be positive it needs to change the8ign to negative. In the process, the top right balancing tree has two edge sign changesto reach balanced state, illustrated by positive-to-negative orange − sign and negative-to-positive teal + sign. Figure 3 illustrates all possible balanced graphs for sample underlyinggraph Σ. Algorithm 1 produced 4 out of 8 possible balanced states, as shown in Figure 5.Not every balanced signed graph is obtainable by balancing algorithm that uses spanningtrees, only the nearest ones are found . Theorem 2.6. If Σ = ( G, σ ) is the signed graph of graph G , then the tree-balancing algorithm(Alg. 1 produces a minimal balancing set for Σ .Proof. Let Σ be the signed graph of graph G ; T, T ∈ T a spanning tree of Σ; and B T bethe balancing set produced by the Tree-Balancing Algorithm 1. If B T is not minimal, thenthere exists a smaller balancing set S ⊂ B T and an element e ∈ B T ∖ S whose reversal is notnecessary to balance Σ. T ∪ e has a unique fundamental circle by design. If e is changingsign, the original one must be negative. That implies e ∈ B T ∖ S , and not reversing e willleave a negative circle. In conclusion, B T must be minimal.If Σ is a balanced signed graph, then F r ( Σ ) =
0, and its minimal balancing set is empty.Theorem 2.6 allows for the re-interpretation of Lemma 2.3 as Lemma 2.7.
Lemma 2.7.
Let Σ be a signed graph, T, T ∈ T be the spanning tree, and B T be the balancingset obtained by T . If B T is balanced tree obtained using Tree-Balancing Algorithm 1, then F r ( Σ ) = min T ∈T ∣ B T ∣ . Figure 5: 8 balanced graphs for signed graph in Figure 4. The 4 in bold can be reconstructed usingtree-balancing algorithm (Alg. 1.
All balanced states of underlying signed graph represent different views of graph consen-sus, and all of them have 0 frustration, per Theorem 2.4. Frustration index (Lemma 2.3)characterizes signed graph as the smallest number of edge sign switches so the signed graphachieves balanced state. If a frustration index of signed graph Σ is
F r ( Σ ) , that means thatΣ is F r ( Σ ) many sign changes from being balanced. In this section we generalize the notion9f the frustration index to the frustration cloud , see Definition 2.1. For a signed graph Σ andits underlying graph G , Σ = ( G, σ ) , the set of edge-signing functions {+ , −} E forms a Booleanlattice L . The all positive edge signing ( G, +) is the element, while the all negative edgesigning ( G, −) are the the element. As we travel further from an ordinary (all positivegraph) the OR operation ∨ produces negative edges, and AND operation ∧ produces positiveedges. Definition 2.1.
Frustration Cloud , F Σ , is the set of all nearest balanced states of a signedgraph Σ . Theorem 2.8.
Let Σ be a signed graph, and let Σ ′ be the balanced state of Σ . Σ ′ is anearest balanced state, that is Σ ′ ∈ F Σ if, and only if, Σ ′ can be obtained by the minimal setof edge-sign inversions.Proof. If Σ ′ ∈ F Σ , it is obtained by a shortest path from Σ to Σ ′ in L as defined in Defn. 2.1.By minimality, there is no subset of edges where a sign-inversion can produce a balancedsigned graph.Tree-balancing algorithm (Alg. 1) produces balanced states with minimal edge-sign in-versions. The set of spanning trees T covers the nearest balanced states, see example inFigure 4. Theorem 2.9.
Let Σ be a signed graph and Σ ′ be a balanced state. Σ ′ ∈ F Σ if, and only if, Σ ′ a result of Algorithm 1.Proof. From Lemma 2.6 and Theorem 2.8, Algorithm 1 produces a minimal balancing setand the resulting F Σ . Now let Σ ′ ∈ F Σ with corresponding minimal balancing set B . Observethat G ∖ B is connected, and any spanning tree in G ∖ B will also be spanning in G . Thus, B is obtained by a spanning tree in G ∖ B . ⇒⇒ Figure 6: Boolean lattice for a signed triangle graph (left); black boxes mark the balanced signedgraphs (center); if underlying signed graph Σ is in green circle, green boxes mark a frustrationcloud F Σ (right). There are eight possible signings of the triangle graph, see Figure 1. Figure 6 (left)illustrates the corresponding Boolean lattice and Hasse diagram: each signed graph is an10lement of the Boolean lattice, with the covering relation that changes exactly one edgenegative via ∨ . All triangle signed graphs have exactly one fundamental cycle. Four of themmeet the balanced state requirement that all cycles are positive, as marked with black boxesin Figure 6 (center). Let signed graph Σ be the graph with two positive edges and onenegative edge, as illustrated in Figure 6 (right) with green circle around the graph. The setof nearest balanced states for Σ is marked with green boxes. Only three out of all possiblebalanced states can be achieved by changing one edge to achieve nearest balanced state,Figure 6 (right). The balanced state in the black box in Figure 6 (right) is not a nearestbalanced state as it can only be reached in the Hasse diagram travelling through one of theother three balanced states. All balanced states provide a consensus-based alternative tothe network. Frustration cloud describes a subset of all balanced states of the graph thatrequire minimal amount of sentiment changes to achieve balance, as illustrated in Figure 6. ⇒ e e e e e ⇒ ∅ Figure 7: Left: The underlying graph G from Figure 4. Middle: The Hasse diagram of all signingsof G signed graph (open square) and balanced states (closed squares) from Figure 4. Right: Thefour elements of F Σ and their shortest paths to Σ from Figure 5(bold). Hasse diagram for the signed graph in Figure 4 is illustrated in Figure 7. It has 5 edges,and there can be 2 different signed graphs on the underlying graph. Figure 4 representvertices not as signed graphs but as an ordered set where edges are labeled by negativeedges, and line segment switches exactly one edge: upward to negative, and downward topositive. The bottom -element is the all-positive signed graph represented by the emptyset as there are no negative edges, while the top -element is the all-negative graph labeledby the entire edge set, Figure 7 (middle). The location of Σ from Figure 4 is the open squareelement, and all balanced graphs from Figure 3 are the closed square elements in Figure 7(middle). All signed graphs that are not in the frustration cloud have been greyed in Figure7 (right), and the shortest paths to the elements of the frustration cloud are indicated inbold.Figure 7 example illustrates the expanded definition of the frustration cloud as there aretwo different balanced states with shortest path 1, and two more minimal paths of lengthtwo to balanced states for Σ. F r ( Σ ) = not in the frustration cloud must pass through one ofelements in the frustration cloud, as illustrated in Figure 7 (right). Balanced states providea consensus-based alternative to the network; frustration cloud is a subset that requires the11east amount of sentiment changes to achieve, and represent the consensus based states thatwill be achieved first. ⇒ Figure 8: Tree-balancing on signed graph Σ produces 4 balanced states. Different spanning treescan produce same balanced state. The edges outside each spanning tree are indicated as dashedlines.
Consensus for social networks is defined in [20] as community resolution when opposingparties set aside their differences and barely agree on a statement. Consensus modeling insocial network analysis has focused on local agreement [14, 16]. There can be multiple waysto achieve global consensus in signed graph: we formalize the consensus modeling throughbalancing theory and the notion of frustration cloud to measure the consensus in a signedgraph. Lemma 2.7 states that a set of spanning trees is the set of bases for the balanced statesof a signed graph. Each tree, coupled with the signs in the original signed graph uniquelyreconstructs a balanced sign graph [35, 52]. Thus, not only is consensus determined byspanning trees, but the nearest consensus is determined by the spanning trees. Differenttree inputs to the Tree-balancing Algorithm 1 can result in the same nearest balanced state,as illustrated in Figure 4. This is a reflection of the underlying network itself, and simplymeans there are multiple ways in which a minimum set of sentiments can be changed toresult in identical outcomes of consensus. Thus, we weight each element of the frustrationcloud by the number of times it is produced by a basis.
Definition 3.1.
For a signed graph Σ = ( G, σ ) , and balanced signed graph Σ ′ ∈ L , the w Σ ′ isdefined as weight of Σ ′ relative to Σ , and it equals the number of spanning trees of G that balance Σ into Σ ′ . An unbalanced signed graph is always assigned a weight of 0, as illustrated in Figure 8.The balanced signed graphs in Figure 8 have weights equal to 3, 3, 1, and 1, as indicatedby the groupings of the balanced states using Algorithm 1. The weight assigns a level of importance to balanced state, as it can be achieved using different underlying spanning trees.12
Figure 9: Harary cuts per balanced state: The deletion of the negative edges in each balanced statein the frustration cloud of the signed graph in Figure 8.
For T G set of spanning trees of a graph G , signed graph Σ = ( G, σ ) and spanning tree T ∈ T G ,Σ ′ T is the balanced signed graph obtained by the Tree-balancing Algorithm 1. The bipartition ( U T , W T ) induced by the Harary-cut in each Σ ′ ( T ) results in two subgraphs, as illustrated inFigure 9, Σ ′ U T and Σ ′ W T . Subgraphs are named so that the following holds ∣ U T ∣ ≤ ∣ W T ∣ : Σ ′ W T isalways the one containing higher or equal number of nodes than Σ ′ U T . Status (Defn. 3.2) canbe regarded as the likelihood that a majority of the vertices in a network can be convincedto agree with a specific node’s position over all nearest balanced states, with multiplicitydetermined by the weight. Figure 10 uses components of Figure 9 are used to determine thestatus.
Definition 3.2.
The status of a vertex v in Σ = ( G, σ ) is defined as the normalized sum ofstep functions if vertex v is in the larger subgraph Σ ′ W T : status ( v ) = ∣T G ∣ ∑ T ∈T G δ Σ ′ WT ( v ) , where δ Σ ′ WT ( v ) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ if v ∈ Σ ′ W T . ∣ W T ∣ = ∣ U T ∣ tie-break otherwise. ⇒ .
58 5 . .
58 3 . Figure 10: Harary-cut for the nearest balanced states weighted by their occurrence (left) and thecalculated status values (clock-wise) for signed graph Σ.
Lemma 3.1.
Let Σ = ( G, σ ) be a signed graph with frustration cloud F Σ . Status can bedefined as a sum of step function if vertex v is in larger balanced sub-graph Σ ′ W T , weightedby w Σ ′ , the number of spanning trees of G that balance Σ into Σ ′ . status ( v ) = ∣T G ∣ ∑ Σ ′ ∈F Σ w Σ ′ δ Σ ′ WT ( v ) . roof. w Σ ′ (Defn. 3.1) trees from Defn. 3.2 result in specific balanced state Σ ′ . Exact sumfrom Defn. 3.2 is contracted along unique balanced states Σ ′ .The top-left vertex of Figure 10 has status (clockwise from top-left group): [ ( ) + ( ) + ( . ) + ( . )]/ = . /
8; the bottom-left is the same: 6 . /
8, top right: [ ( ) + ( . ) + ( )]/ = . . /
8; bottom-right: [ ( ) + ( . ) + ( . )]/ = . / sum of all statuses of all vertices in a graph, and Definition 3.2. Lemma 3.2.
For signed graph Σ = ( G, σ ) , vertex set V , and tree-spanning set T G , the sumof statuses of all vertices in Σ equals normalized sum of cardinality of larger component ofthe Harary-cut over all spanning trees T, T ∈ T G . ∣T G ∣ ∑ v ∈ V status ( v ) = ∑ T ∈T G ∣ V ( Σ ′ W T )∣ . Proof.
Summing both sides of the definition of status over all vertices gives ∑ v ∈ V status ( v ) = ∑ v ∈ V ∣T G ∣ ∑ T ∈T G δ Σ ′ WT ( v ) = ∣T G ∣ ∑ v ∈ V ∑ T ∈T G δ Σ ′ WT ( v )= ∣T G ∣ ∑ T ∈T G ∑ v ∈ V δ Σ ′ WT ( v ) = ∣T G ∣ ∑ T ∈T G ∣ V ( Σ ′ W T ))∣ . The last equality holds even in the event of having two components of equal size, aswe have defined status. Since δ Σ ′ WT ( v ) treats them as equal split (0 .
5) there are the samenumber of vertices in both the new majority as well as the minority — which is equivalentto counting the size of the tied majority. The proof is completed by multiplying by ∣T G ∣ . Consider the variation of attitudinal strength captured by edge sign in attitudinal networks.When students assign a strong rating score for instructor evaluation, it is hard to separateaffective, behavioral, and cognitive components of the attitude expressed in that one senti-ment. Did the student take all of their other instructors into consideration? What is thesubjective evaluation range? How much of the rating is based on students’ own subjectiveperformance in the class? How likely is the student to change his mind when he talks to hispeers? We do not have the answer to any of these questions. All we have is edge sentiment,and the ability to quantify its strength and reliability of that sentiment within network.Tree-based balancing algorithm 1 changes signs of the edges outside of underlying spanningtree to achieve balancing, see Figure 8. We propose a new measure, termed agreement tomeasure how indicative a single edge is on true vote. It is the likelihood that an edge willbe positive, and contribute to the consensus decision in all near balanced states producedby tree balancing algorithm, see Definition 3.3.14 efinition 3.3.
The agreement of an edge e in a signed graph is a normalized sum of alloccurrences of an edge in the largest component of a Harary-cut over all spanning trees. agreement ( e ) = ∣T G ∣ ∑ T ∈T G δ Σ ′ WT ( e ) , where δ Σ ′ WT ( e ) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ if e ∈ Σ ′ W T . ∣ W T ∣ = ∣ U T ∣ tie-break otherwise. Figure 11 shows agreement for given example. The larger agreement an edge has, themore likely it will appear in the final, majority decision. .
58 183 . .
58 28 ⇒ Figure 11: Left: Harary-cut; Right: Edge Agreement values.
Figure 10 illustrates that the bottom-left and top-left vertices both have the largest statusvalues. Agreement helps quantify the differences between them. The influence in the graphof top-left vertex is higher than bottom-left vertex, as the total agreement for all the edgesoriginating in top left vertex higher . Lemma 3.3.
For signed graph Σ = ( G, σ ) , edge set E , and tree-spanning set T G , the sumof agreement of all edges e, e ∈ E , in Σ equals normalized sum of edge cardinality of largercomponent of the Harary-cut over all spanning trees T, T ∈ T G . ∣T G ∣ ∑ e ∈ E agreement ( e ) = ∑ T ∈T G ∣ E ( Σ ′ W T )∣ . Parallel to Lemma 3.2, and following the definition 3.3 of individual agreement, Lemma 3.3states that sum of all agreements over all edges is constant. Edge agreement is defined inDefn. 3.3, and vertex agreement, influence , is (normalized) sum of all edge agreements pernode, as defined in Defn. 3.4.
Definition 3.4.
The influence of a vertex v in a signed graph is the average agreement ofall edges incidence to vertex v , inf luence ( v ) = deg ( v ) ∑ e ∼ v agreement ( e ) . .
58 183 . .
58 28 6 .
58 5 . .
58 3 . ⇒ .
338 3 . .
58 2 . Figure 12: Signed graph Σ in Figure 4, and its edge agreement values (left); vertex influence(center); and vertex status (right).
Status and influence provide two measures of vertex influence in the attitudinal networkgraph, as illustrated in Figure 12. Relationship of status and influence measures for vertex v in outlined in Lemma 3.4. Their relation stems from Defn. 3.2 and Defn. 3.4, and fromcomparing the totality of edge counts around each vertex. Lemma 3.4.
For a signed graph inf luence ( v ) < status ( v ) . Moreover, equality holds when v is a pendant vertex whose edge is positive; and influence is when v is a pendant vertexwhose edge is negative. Consensus is a general agreement that can be achieved without unanimous voting. If consen-sus in the signed graph is unanimous, then the Harary-cut produces one partition consistingof the entire connected graph: the nearest balanced state has all positive edges. On theother end of the spectrum, nearest balanced state can result in a Harary-cut that has bi-partitions of equal size, and entire graph is deadlocked in indecision. Controversy (fromlatin controversia , turn in opposite direction) occurs anytime there are conflicting opinionsin the group. Controversy in balanced graph states occurs when consensus is achieved butthe voting was not unanimous. Each balanced state but one (all positive edge) has a certainlevel of controversy associated with it. The measure of average status of the nearest balancedstates can quantify controversy for underlining signed graph, and graph status definition isDefinition 3.5.
Definition 3.5.
Let status ( Σ ) denote the graph status measure, and ∣ V ( G )∣ is the numberof vertices in the graph. Then, an average status of a signed graph Σ is defined as status ( Σ ) = ∣ V ( G )∣ ∑ v ∈ V status ( v ) Lemma 3.5.
Let Σ = ( G, σ ) be a signed graph, then . ≤ status ( Σ ) ≤ .Proof. This lemma sets the bounds of status sum in Lemma 3.2. From the definition ofmajority we have that for every spanning tree T we have ∣ V ( G )∣ ≤ ∣ V ( Σ ′ W T )∣ ≤ ∣ V ( G )∣ . ∣T G ∣ ∑ v ∈ V status ( v ) is a sum of the sizes of the majority, so itmust be an integer. The bounds are from Theorem 3.5 and multiplying by ∣T G ∣ . CombiningLemmas 3.2 and 3.5 we have: Theorem 3.6.
For a signed graph Σ = ( G, σ ) and for all spanning trees T of Σ :1. status ( Σ ) is minimal ( = . ) if, and only if, ∣ V ( Σ ′ W T )∣ = ∣ V ( Σ ′ U T )∣ , ∀ T status ( Σ ) is maximal ( = ) if, and only if, ∣ V ( Σ ′ W T )∣ = ∣ V ( G )∣ , ∀ T Average status over all vertices in the graph is a measurement of controversy (Theorem3.6 ). Maximum value of status ( Σ ) = .
0: the case when all nearest balanced states haveall positive edges, and everyone agrees all the time. Minimum value of status ( Σ ) = .
5, thebalancing consistently splits the set in two equally sized subsets. In between, if status ( Σ ) iscloser to 1, the entire graph has low controversy, and if it is trending to 0 .
5, the entire graphas higher controversy. Lets expand the notion of tie-break from definition 3.2 of status.One way to resolve a tie-break in Section 3.1 is to assign status and agreement values of0.5 if the Harary-cut bipartitions are equal size. In the Human Resource Scenario, considerthe scenario where ”reliable” or ”reputable” vertex exists, and have that person (vertex insigned graph) breaks ties in its own favor when a Harary-cut bipartition are of equal size.Lets define vertical status in Definition 3.6.
Definition 3.6.
The vertical status of a vertex v in Σ = ( G, σ ) with respect to designatedvertex t is status t ( v ) = ∣T G ∣ ∑ T ∈T G δ t Σ ′ WT ( v ) , δ t Σ ′ WT ( v ) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ if v ∈ Σ ′ W T , ∣ W T ∣ = ∣ U T ∣ , and v, t in the same partition, otherwise. Definition 3.6 states that all tie-breaks increase the status of vertex t and vertices in thesame subset as t . Figure 13 illustrates how vertical status compares to status for signedgraph from Figure 4. Lets consider the top-left vertex (now a closed box) as the tie-breakerFigure 13 in case 1, and the bottom-right vertex (now an open box) the tie-breaker in case 2. Case 1 : the top-left vertex is used to break any ties, and the vertical status values are now ( / , / , / , / ) , as illustrated in Figure 13(middle). Case 2 : the bottom-right vertex isused to break ties, and the vertical status values are now ( / , / , / , / ) . In both cases,the status of the chosen vertex increased over the original status in Figure 10.17
58 4888 5888 7858 28
Figure 13: Calculating the vertical status using the top-left (middle) and bottom-right (right)vertex.
Lemma 3.7.
For a signed graph Σ = ( G, σ ) and vertex v ∈ V , the status t ( v ) is maximizedwhen t = v .Proof. If t = v , vertex determines its own tie-breakers, and every 0 . δ Σ ′ WT ( v ) is replacedwith a 1 in δ t Σ ′ WT ( v ) . Definition 3.7.
Let status t ( Σ ) denote the average vertical status of a signed graph Σ withdistinguished vertex t ∈ V . status t ( Σ ) = ∣ V ( G )∣ ∑ v ∈ V status t ( v ) Theorem 3.8 (Conservation of Controversy Law:) . For a signed graph Σ = ( G, σ ) , graphcontroversy is equal to its status and any vertical status: controversy ( Σ ) = status ( Σ ) = status t ( Σ ) , ∀ t ∈ G Proof.
As in Theorem 3.2, ∑ v ∈ V status t ( v ) = ∑ v ∈ V ∣T G ∣ ∑ T ∈T G δ t Σ ′ WT ( v ) = ∣T G ∣ ∑ v ∈ V ∑ T ∈T G δ t Σ ′ WT ( v )= ∣T G ∣ ∑ T ∈T G ∑ v ∈ V δ t Σ ′ WT ( v ) = ∣T G ∣ ∑ T ∈T G ∣ V ( Σ ′ W T )∣ = ∑ v ∈ V status ( v ) . The second to last equality is is a strict count of the size of the majority, while the lastequality is from Lemma 3.2. The proof is completed by dividing by ∣ V ∣ . Conservation of Controversy Law in Theorem 3.8 states that the average statusis equal to any average vertical status. Interpreting average status as controversy, we canconclude that the level of controversy is independent of vertex preference , but the individualstatus values may change. Conservation of Controversy Law: status in Figure 10(right) is0 . . . Consensus through Frustration Cloud in Practice
Proposed measures and definitions of status 3.2, agreement 3.3, and controversy 3.6 of thesigned graph require computation of all spanning trees for underlying unsigned graph G .Attitudinal Network Graphs can have from few thousand to few million vertices, and numberof edges [26, 30] makes them likely sparse. The social ANGs are inherently sparse i.e. thenumber of vertices is n , the number of links is usually smaller than n ∗ ( n − )/
2, a criteriafor sparsity.
Tutte’s reclaiming of Kirchhoff’s laws via graph theory [7, 15, 41, 48] computes the num-ber of nonidentical spanning trees of a graph G is the determinant of any first minor ofthe Laplacian matrix of G . This is the extension of Cayley’s theorem that complete graphwith n vertices has n n − spanning trees, and a complete bipartite graph with m, n verticeshas m n − n m − spanning trees. The Highland Tribe Data has 16 vertices and 58 edges [39](see Figure 21), and 402 , , ,
163 nonidentical spanning trees. That is computationallyprohibitive, and a large number of the spanning tree graph discovery and sampling meth-ods has been proposed [42]. Approximate modeling of balance state coverage (frustrationcloud), and balanced weighing when computing status, influence, and controversy dependson spanning tree discovery approach. Law of Conservation of Controversy (Theorem 3.8)holds for any fixed subset of spanning trees. Different subsets of trees may produce differentcontroversy values. We examine random, breadth-first, and depth-first spanning tree sam-pling and demonstrate how status, influence, and controversy can be perceived in differentparadigms.
Random Sampling : Uniform spanning tree is a tree chosen randomly from among allthe spanning trees with equal probability. Wilson’s algorithm takes a random walk on graph G and erases the cycles created by this walk [50]. This algorithm can be used to generateuniform spanning trees in polynomial time. Random Minimal Spanning tree algorithm gen-erates random trees, but cannot guarantee uniformity. The edges of the graph are assignedrandom weights and then the minimum spanning tree of the weighted graph is constructed.We use implementation of three different algorithms in NetworkX [17] for discovery of min-imal Spanning tree algorithms, namely DJP, Boruvka, and Kruskal’s algorithm [17, 42].They are all greedy algorithms that run in polynomial time, and we have not observed muchdifference in speed or randomization when selecting specific one. Our assumption here wasthat a random sample will best represent frustration cloud and multiplicity. Random span-ning trees provide a straight-forward way to analyze the frustration cloud, as context-drivenalgorithms such as breadth- or depth-first alter the resolution of the data.
Breadth-first search (BFS) is an algorithm for spanning tree search in a network graphthat progressively explores all the neighborhood vertices from starting vertex (search key)at present depth before moving to next depth level. As a result, breadth-first spanning treesare the most star-link trees in the graph. Lovasz eigenvalues of trees theorem demonstratesthat stars represent the classes of tress that have maximal eigenvalues [32], and breath-first19panning tree search maximizes the sampling of star-like trees. Breath-first search algorithmhas been used to find the shortest path two vertices in a graph measured by number of edges[42], this allows for the shortest cycles for feedback in an attitudinal network. Spanning treesresulting from breath-first search have maximal number of pendant vertices and fundamentalcycles of minimal length [42].
Depth-first search (DFS) algorithm is complementary to breath-first search discoveryof spanning trees as it explores the branch to the highest depth level possible, before back-tracking and expanding [42], effectively delaying cycle feedback as long as possible. Treesdetermined by depth-first a approach are the most path-like trees with minimal number ofpendant vertices. The process maximizes the length of the fundamental cycle, and depthfirst search algorithm has been used in determining the number of connected componentsin a graph and topological storing [42]. Lovasz’ eigenvalues of trees theorem demonstratesthat paths represent the classes of trees that have minimal eigenvalues [32], and depth-firstspanning tree search maximizes the sampling of path-like trees.Larger eigenvalues dominate convergence, and coupled with the minimization of funda-mental cycle length by breath-first trees, we conjecture that breadth-first tree sub-samplingwill provide the greatest data resolution for status, while in depth-first tree sub-samplingwill produce a noisy interpretation of the same data. We demonstrate the validity of thisassumption on larger Wikipedia administrator dataset in Section 5.1.
The computation of frustration cloud-based measures for signed graph is implemented usingPython, and Python libraries, and the code used for proof-of-concept is set to be releasedon github [47] in 2020. The analysis of largest connected component, spanning tree searchmethods, and statistics was computed using NetworkX [17] package. Experiments were runon Texas State University LEAP system [12]: Dell PowerEdge C6320 cluster node consists oftwo (14-core) 2.4 GHz E5-2680v4 processors 128 GB of memory each, and two 1.5TB memoryvertices with four (18-core) 2.4 GHz E7-8867v4 Intel Xeon processors [12]. LEAP systemallowed us to scale the data analysis and support the sampling of n = graphB pipeline, spanning trees are generated over the dataset, and saved in h5format: we use NetworkX [17] implementation of random minimal tree, breadth-first, anddepth-first tree discovery. Next, for each generated spanning tree, the balancing algorithm isexecuted on the edges not in the spanning tree, see Alg. 1 for more details. We obtain a listof unique paths encompassing the spanning tree and given edge, and check for fundamentalcycles. If the product of the cycles is −
1, then the given edge completing the cycle changingsigns, resulting in balanced cycle. We repeat the process for each edge. Once all edgesoutside the spanning tree are visited and edge signs persisted or flipped, the resulting stateis a balanced state of the graph. Next, we take a Harary-cut, and split the graph into twocomponents, and compute status for each vertex and agreement for each edge as normalizedsum over sampled 1000 trees, per Defn. 3.2 and Defn. 3.3. Vertex influence is then computedas normalized sum over the sampled 1000 trees per Defn 3.4, and controversy of the entire20raph per Theorem 3.6.
Stanford Network Analysis Project’s (SNAP) repository of network data provides good proof-of-concept access to attitudinal network graphs [30].
Wikipedia administrator election datarepresents votes by Wikipedia users in elections for promoting individuals to the role ofadministrators from July 2004 to January 2008. Wikipedia administrators are editors whohave been granted the ability to perform special tasks. The dataset contains 7118 users(vertices) and 103,747 votes (edges) over 2794 elections, one election per candidate, andoutcome of the elections. Out of 2794 elections, 1235 resulted in promotion to administrator(44 . communityreview process that seeks consensus not a majority rule, as editor in charge reviews editors’votes and rationale. Neutral votes are not included in the calculation but the rationale arestill considered to assess consensus. Attitudinal graph from Wikipedia administrator electiondata is constructed so that each vertex represents editor or nominee: if both or runningfor multiple administrator position (possible, as there are different Wikipedia sections), werepresent one user with multiple vertices, where each vertex has a final outcome (winner,loser, editor). The edge in the graph models the vote of support or initial nomination ( + − N spanning trees and balanced states, status and influence are computed as inDefn. 3.2 and Defn. 3.4 as following: Definition 5.1.
For signed graph Σ = ( G, σ ) , tree-spanning set T G , its T N , ∣T N ∣ = N , statusof vertex v, v ∈ V and agreement of edge e, e ∈ E , are computed as: status ( v ) = N ∑ T ∈T N δ Σ ′ WT ( v ) , where δ Σ ′ WT ( v ) is defined in Defn. 3.2, agreement ( e ) = N ∑ T ∈T N δ Σ ′ WT ( e ) where δ Σ ′ WT ( e ) is defined in Defn. 3.3, inf luence ( v ) = deg ( v ) ∑ e ∼ v agreement ( e ) , controversy ( Σ ) = ∣ V ∣ ∑ v ∈ V status ( v ) . Connected components: . . N spanning trees, N is in ( , , ) set. Measure of status and influence for three different spanning tree discovery techniques onWikipedia dataset for N = random trees as determined by21inimal spanning trees with random edge weights; breadth-first trees with a random initialvertex; and depth-first trees with a random initial vertex. Results for status and influencefor wiki data editors and nominees for three spanning tree discovery techniques are shown inFigure 14. The conjecture from Section 4.1 is shown valid in practice, as breath-first searchprovides highest resolution of status and influence metric analysis. Figure 14: Status (first three figures) and influence (second three figures) of editors (gray dots)and nominees (yellow dots) in Wikipedia administrative election dataset resulting from differenttree sampling methods: Random minimal spanning tree (left), breath-first (center), and depth-first(right).Figure 15: Status (first three figures) and influence (second three figures) of winners (blue dots)and losers (red dots) in Wikipedia administrative election dataset resulting from different spanningtree discovery methods: Random minimal spanning tree (left), breath-first (center), and depth-first(right).
The influence of gray dots (editors) is high in Figure 14’s right three figures over averageinfluence of nominees, and it justifies influence as a metric for ability to enact change inthe network (i.e. promote someone). Figure 14 bands are mean and standard deviationfor average editor and status measures for all three discovery methods. Status does notdiscriminate between voters and nominees in Figure 14, and influence measure does as highinfluence individuals are mainly voters. Analysis of only nominees in the light of the outcomesis in Figure 15: blue is a positive outcome (promoted to administrator), red is a negativeoutcome (not promoted or withdrew its nomination). Figures 15 captures different measuresinfluence and status present: high status individuals are mostly the nominees that won theadministrator election, and low status individuals are mostly the nominees that did not winthe election. Sensitivity of status measure for nominees and its strong relation to outcomemakes status a good predictor of promotability in random and breath-first spanning treediscoveries. Depth-first experiment consistently produces biased sample in terms of balancedstates and frustration cloud, and our experiments on other datasets consistently confirm theconjecture [47]. 22 able 1: Spanning three discovery methods comparison on wiki data:
Type Mean Status (Controversy) St.Dev status Mean Influence St. Dev influencebreath-first while depth-first value is . Controversy, as defined in Theorem 3.6, isa constant value when all spanning trees are accounted for, and breath-first and depth-firstgive us the range estimate of the value. Controversy is relative measure of attitudinal net-work graph, and if compared to another graph, same tree discovery and sampling methodmust be used for valid comparison.
What is the sufficient number of spanning trees ( N in Defn. 5.1) that will produce reliablemetric for status and influence? Here, we access the sensitivity of status and influence tonumber of spanning trees sampled for random minimal tree and breath-first search spanningtree for N , N = N = N = Figure 16: Sensitivity of random (first row) and breath-first (second row) search tree sampling tonumber of sampled trees N , N = N =
100 and N = N =
10 status can only take one of10 different values (is the vertex in majority or not for each sampled balanced state), asillustrated in Figure 16 (right). Shelving effect is visible. Influence, based on Defn. 3.4 has ahigher resolution, as it is based on average of agreement for each vertex, and vertex measurediffers. Shelving effect is visible for a tight group of editors that demonstrate identicalbehavior in a small sample of balanced states Figure 16 (right) for both random and breath-first sampling. Higher N allows for more diverse samples to contribute to status, and whilethe overall resolution of both discovery strategies is smaller, it provides for better results.Figure 16 shows that the status values have the higher resolution, and lower number of treesachieves similar separation results in nominee outcome status as higher number of trees,similar behavior as random sampling. What is the guiding principle for larger attitudinalgraphs? We have tested the method on larger signed graph Slashdot, and N = Measures of status and influence can be used to access the outcome for a vertex in attitudinalgraph. Requests for adminship (RfA) is the process by which the Wikipedia communitydecides to promote nominees into administrators [13]. Here, we use
RfA as a measure oftotal votes for the nominee : normalized sum of all edges originating in nominee vertex v : Rf A ( v ) = deg ( v ) ∑ e ∼ v e . Election outcomes are blue dots are candidates that won theelection, and red dots are candidates that lost the election. Nominee status is obtained byUser submitting own requests for adminship or being nominated by editors. Figure 17: Wikipedia data analysis of the request for adminship (RfA) [13] process from status andinfluence perspective. Blue dots are users that won the election, and red dots are users that lostthe election: RfA vs. user id (left); status vs RfA (center); influence vs. RfA (right).
Final outcome is a complex process that involves majority voting (RfA in Figure 17 andvetting process. In general, if number of positive votes is under 65% nominee is rejected, andif number of votes is over 75%, nominee is selected. Vetting process and discussion determinethe final outcome. Burke proposed a model of the behavior of candidates for promotion toadministrator status in Wikipedia [8]. He analyzes multiple measurable features of thenominee (strong edit history, varied experience, user interaction, helping with scores) andhighlights similarities and differences in the communityâĂŹs stated criteria for promotiondecisions to those criteria actually correlated with promotion success. Here, we propose touse status and influence as features of consensus in the entire network. Status and influence24o not consider candidates features, but solely its position in attitudinal network, and itsrelationship to RfA is illustrated in Figure 17.
Table 2: Measure distribution in Wiki adminship dataset for N = Mean (St.Dev) Nominees Promoted Not PromotedRfA 0.9476 (0.0742) 0.3055 (0.2826)Status 0.6097 (0.2962) 0.8632 (0.0719) 0.4009 (0.2433)Influence 0.4414 (0.2579) 0.6721 (0.0803) 0.2514 (0.1898)Mean (St.Dev) All EditorsRfA 0.5958 (0.3852) N/AStatus 0.6693 (0.2564) 0.7041 (0.2226)Influence 0.5178 (0.2823) 0.5624 (0.2864)
Figure 18: Wikipedia data analysis of the voting process from status vs. influence perspective, leftto right: editors (gray) vs. nominees (yellow); RfA score for nominees from yellow to blue; RfAdecision bands for nominees: red is the range of RfA scores where it is up to editors to make adecision; and Wikipedia outcome for nominees: blue - elected; red - rejected.
We identify editors from Figure 18(left) gray dots with highest status and influence asleaders in the opinion swaying. The method analyzes Wikipedia election majority votingresults (RfA) and final outcome wrt to status and influence measure. We flag spam users,privileged users, narrow domain users and all anomalies in the process using simple rules:(1) for all RfA in [65,75) % range (red dots in third image in Figure 18), we flag electionoutcomes for users with low status and low influence that got elected (blue dots in the lowerleft quadrant of fourth image in Figure 18) and users with high status and high influencethat were not elected ((red dots in the upper right quadrant of fourth image in Figure 18).For example our algorithm uncovered several interesting cases: ID had status 0 . . bozmo had status 0 . . tjstrf had status 0 .
905 and influence 0 . mn had a status 0 .
448 and influence 0 . Slashdot Zoo is a signed social network has 82 ,
144 users (vertices) and 549,202 edges, 77 . ,
052 (99 . Figure 19: Slashdot friend-foe network analysis using frustration cloud approach and N = We construct attitudinal graph from friend-foe relationship, and analyze the status andinfluence of users. Results are presented in Figure 19, with frustration cloud and N = Frustration cloud based approach allows for a more robust way to analyze the perceivedoutcome in attitudinal network graph, as it is based on mathematical sociology model forbalanced system. Highland tribes datasets captures the alliance structure of a network oftribes of the Eastern Central Highlands of New Guinea [39]. The network contains sixteentribes (vertices), and edges represent agreement (”rova”) or animosity (”hina”) betweentwo tribes, as illustrated in Figure 20 with solid lines for agreement, and dashed lines foranimosity. Read’s ethnography portrayed an alliance structure among three tribal groupscontaining balance as a special case, as the enemy of an enemy can be either a friend or anenemy [36]. There are 16 vertices (tribes) and 58 signed edges (tribe relations): 29 positive(sign +1) and 29 negative (sign -1). Signed graph Σ for the Highland tribes dataset isconstructing by adding two provided matrices. The edges in the graph reflect positive (sign+1) or negative (sign -1) relationship between two tribes, as illustrated in Figure 21(left).Highland Tribe Data has 16 vertices and 58 edges and has 402 , , ,
163 spanning trees.We sample 1000 spanning trees using breath-first approach for this experiment. Highlandtribes relation in Figure 20 separates two groups of tribes. The gray shade and size of vertexcircle corresponds to computed status per Defn. 3.2 and Defn. 3.6.
Figure 20: Highland Tribe Status computation for breath-first sampled 1000 trees: solid circles aretribes, solid lines are agreeable relations, dashed lines are antagonistic relations between two tribes,the size of the vertex circle illustrates computed status for that circle. Shade of gray scale for the status of the vertex per Defn. 3.2 (left); color coded change of status (center) and status (left)vs. status per Defn. 3.6, where blue is an increased status and red is a decreased status. Status is computed as normalized number of times over N = status , status ,27nd status ), as explained in Section 3.3. Vertices 0, 1, 14, and 15 form a smaller group,and it is reflected in the lowest status scores for those 4 vertices and lower overall status.Corresponding temperature graph show how the vertical status maximizes status for theselected vertex and connected vertices Figure 20. Figure 20(center) illustrates the change instatus when tie-break node is from the smaller cluster. difference of vertical status (3.6) and status (Defn. 3.2). While the status of all 4 nodes in that community grows significantlyat the expense of the reduced status of vertices in the majority group. Figure 20(right)illustrates the maximization of the status in the majority group if vertex 6 is selected as atie breaker, vertical status (3.6) and status (Defn. 3.2). Figure 21: Highland Tribes relationship graph (left): solid circles are tribes, solid lines are agreeablerelations, while dashed lines are antagonistic relations between two tribes. The size and share of thecircle represents computed status per Defn. 3.2; vertex influence computed per Defn 3.4(center);status vs. influence correlation (right).
Figure 21(left) illustrates the status difference per vertex id for status , status , and status . Solid black line demonstrates Law of Conservancy, as controversy for the highlandgraph is constant at 10 / Figure 21(center) shows influence (Defn. 3.4) as a function of vertex id, and the same 4 nodeshave the lowest computed influence. Influence and status are highly correlated as shown inFigure 21 (right) plots. Interesting observation on nodes 4, 6, and 7 is that they all havehigh status: influence of vertex 4 (
Nagam tribe) is less than its status in the network, andinfluence of vertices 6 and 7 (
Masil and
Ukudz tribes) is higher than its status in the network.graphB analysis provides a simple, unbiased view into Highland data, and flags 3 out of 16tribes to be re-examined deeper by the anthropology experts.28 igure 22: Spectral clustering of Highland Tribes data: only positive edge clustering producesmeaningfully communities, negative information is lost, see results for K=2 and K=3 (left); statusvs. influence measures colored by spectral clustering results for K=2 and K=3 (right).
Figure 22 compares the results of spectral clustering with status and influence mea-surements. Spectral clustering as-is makes sense only on positive relations to understandcommunity grouping, and animocity measure is lost, as shown in Figure 22(left). Statusvs. influence ranking, colored by Spectral clustering for K=2 and K=3 is shown in Fig-ure 22(right). Status and influence already provide deterministic measures for hierarchicalgrouping of the nodes, and its ordering by status and influence is hierarchical and containsfull information on vertex influence in the network. There is no need to specify K like forspectral clustering, as the status and influence rank and group nodes in 2-D space.
In this paper, we propose to model cognitive “correction” in a attitudinal graph networkusing generalized balance theory. The proposed work examines nearest balanced states ofa given network, and models attitudinal strength and influence in the network throughmeasurements of a network’s ability to reach a balanced state with a minimal number ofedge sign changes. We introduce the concept of ”frustration cloud” and multiple measuresthat capture vertex and edge relation to nearest balanced state of the network. We presentthe Conservation law of Controversy that maximizes the status of the deciding vertex whilekeeping network controversy constant. We demonstrate the usefulness of introduced methodsand metrics on Wikipedia, Slashdot, and Highland datasets.We are currently working on metric to measure the strength of vertex and edge interac-tions; measure that utilizes known promotional outcomes to detect and quantify bias for themajority/minority; and a toolkit for deeper analysis of the proposed metrics. On the appli-cation side, we are focusing on collaborative networks, both humans and agents. Consensusmodeling has gained traction as a way to model agreement in multi-agent networks in pres-ence of antagonistic interactions [4, 45]. We plan to apply balancing theory to multi-agentnetwork agreement, and see if we can identify and tune existing policies, and detect biasedagents (AI modules) in the network.
Acknowledgements:
We would like to thank Data Lab students Joshua Mitchell for initialimplementation and graphB 1.0 proof-of-concept, Eric Hull for graphB 2.0 proof-of-concept29nd code release, and Maria Tomasso for providing data analysis and visualization resultsused in the paper. We would like to thank Texas State for its support through startupfunding and computational facilities.
References [1] Robert P. Abelson and Milton J. Rosenberg. Symbolic psycho-logic: A model of atti-tudinal cognition.
Behavioral Science , 3(1):1–13, 1958.[2] M.J. Alava, P.M. Duxbury, C.F. Moukarzel, and H. Rieger. Exact combinatorial algo-rithms: Ground states of disordered systems. In
In: C. Domb and J.L. Lebowitz, eds.,Phase Transitions and Critical Phenomena, Vol. 18 . Academic Press, San Diego, 2001.[3] S. Alemzadeh, M. H. de Badyn, and M. Mesbahi. Controllability and stabilizabilityanalysis of signed consensus networks. In , pages 55–60, Aug 2017.[4] C. Altafini. Consensus problems on networks with antagonistic interactions.
IEEETransactions on Automatic Control , 58(4):935–946, 2013.[5] F. Barahona, R. Maynard, R. Rammal, and J.P. Uhry. Morphology of ground states oftwo-dimensional frustration model.
J. Phys. A: Math. Gen. , 15:673–699, 1982.[6] C. Berge. Sur certains hypergraphes g´en´eralisant les graphes bipartites. In
Combina-torial theory and its applications, I (Proc. Colloq., Balatonf¨ured, 1969) , pages 119–133.North-Holland, Amsterdam, 1970.[7] F. Buekenhout and M. Parker. The number of nets of the regular convex polytopes indimension ≤ Discrete Mathematics , 186(1):69 – 94, 1998.[8] Moira Burke and Robert Kraut. Mopping up: Modeling wikipedia promotion decisions.In
Proceedings of the 2008 ACM Conference on Computer Supported Cooperative Work ,CSCW ’08, pages 27–36. ACM, 2008.[9] D. Cartwright and F. Harary. Structural balance: a generalization of Heider’s theory.
Psychological Rev. , 63:277–293, 1956.[10] Xi Chen, Jefrey Lijffijt, and Tijl De Bie. Quantifying and minimizing risk of conflict insocial networks. In
Proceedings of the 24th ACM SIGKDD International Conference onKnowledge Discovery & , KDD ’18, pages 1197–1205, 2018.[11] James A. Davis. Clustering and structural balance in graphs.
Human Relations ,20(2):181–187, 1967.[12] Texas State University Division of Information Technology. Leap - high performancecomputing cluster. https://doit.txstate.edu/rc/leap.html , 2020.3013] Texas State University Division of Information Technology. Requests for adminship(rfa). https://en.wikipedia.org/wiki/Wikipedia:Requests_for_adminship , 2020.[14] Kiran Garimella, Gianmarco De Francisci Morales, Aristides Gionis, and Michael Math-ioudakis. Reducing controversy by connecting opposing views. In
Proceedings of theTenth ACM International Conference on Web Search and Data Mining , WSDM ’17,pages 81–90, 2017.[15] W. Grilliette, J. Reynes, and L. J. Rusnak. Incidence hypergraphs: Injectivity, unifor-mity, and matrix-tree theorems.
ArXiv:1910.02305 [math.CO] , 2019.[16] R. Guha, Ravi Kumar, Prabhakar Raghavan, and Andrew Tomkins. Propagation oftrust and distrust. In
Proceedings of the 13th International Conference on World WideWeb , WWW ’04, pages 403–412. ACM, 2004.[17] Aric Hagberg, Pieter Swart, and Daniel S Chult. Exploring network structure, dynamics,and function using networkx. https://networkx.github.io/documentation/stable/reference/algorithms/index.html , Jan 2008.[18] F. Harary. On the notion of balance of a signed graph.
Michigan Math. J. , 2(2):143–146,1953.[19] F. Harary. On the measurement of structural balance.
Behavioral Sci. , 4:316–323, 1959.[20] Tim Hartnett.
Consensus-Oriented Decision-Making: The CODM Model for FacilitatingGroups to Widespread Agreement . New Society Publishers, 2011.[21] F. Heider. Attitudes and cognitive organization.
J. Psychology , 21:107–112, 1946.[22] J. Hu and W. X. Zheng. Bipartite consensus for multi-agent systems on directed signednetworks. In , pages 3451–3456, Dec2013.[23] J.E. Hunter, J.E. Danes, and S.H. Cohen.
Mathematical Models of Attitude Change:Change in single attitudes and cognitive structure . Change in Single Attitudes andCognitive Structure. Academic Press, 1984.[24] M.A. Javed, M.S¿ Younis, S. Latif, J. Qadir, and A. Baig. Community detection innetworks: A multidisciplinary review.
Journal of Network and Computer Applications ,108, 2018.[25] Y. Jiang, H. Zhang, and J. Chen. Sign-consensus of linear multi-agent systems oversigned graphs using a fully distributed protocol. In , pages 3537–3541, Dec 2016.[26] Scott P. Kolodziej, Mohsen Aznaveh, Matthew Bullock, Jarrett David, Timothy A.Davis, Matthew Henderson, Yifan Hu, and Read Sandstrom. The suitesparse matrixcollection website interface.
Journal of Open Source Software , 4:1244–1248, March 2019.3127] Srijan Kumar, William L. Hamilton, Jure Leskovec, and Dan Jurafsky. Communityinteraction and conflict on the web. In
Proceedings of the 2018 World Wide Web Con-ference , WWW âĂŹ18, page 933âĂŞ943, Republic and Canton of Geneva, CHE, 2018.International World Wide Web Conferences Steering Committee.[28] Jure Leskovec, Daniel Huttenlocher, and Jon Kleinberg. Predicting positive and negativelinks in online social networks. In
Proceedings of the 19th International Conference onWorld Wide Web , WWW ’10, pages 641–650. ACM, 2010.[29] Jure Leskovec, Daniel Huttenlocher, and Jon Kleinberg. Signed networks in social media.In
Proceedings of the SIGCHI Conference on Human Factors in Computing Systems ,CHI ’10, pages 1361–1370, 2010.[30] Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network datasetcollection. http://snap.stanford.edu/data , June 2014.[31] K W Li, D M Kilgour, and K W Hipel. Status quo analysis in the graph model forconflict resolution.
Journal of the Operational Research Society , 56(6):699–707, 2005.[32] L. LovÃąsz and J. PelikÃąn. On the eigenvalues of trees.
Periodica MathematicaHungarica , 3:175–182, 1973.[33] Linyuan Lu and Tao Zhou. Link prediction in complex networks: A survey.
Physica A:Statistical Mechanics and its Applications , 390(6):1150 – 1170, 2011.[34] Abhinav Mishra and Arnab Bhattacharya. Finding the bias and prestige of nodes in net-works based on trust scores. In
Proceedings of the 20th ACM International Conferenceon World Wide Web (WWW) , pages 567–576, 2011.[35] James G. Oxley.
Matroid theory . Oxford Science Publications. The Clarendon PressOxford University Press, New York, 1992.[36] Hage P. and Harary F.
Structural models in anthropology . S. Cambridge: CambridgeUniversity Press., 1983.[37] L. Pan, H. Shao, and M. Mesbahi. Laplacian dynamics on signed networks. In , pages 891–896, Dec 2016.[38] Nicolas Rashevsky.
Mathematical Theory of Human Relations: An Approach to Math-ematical Biology of Social Phenomena.
Principia Press, Bloomington, ID, 2nd edition,1947/1949.[39] Kenneth Read. Cultures of the central highlands, new guinea.
Southwestern Journal ofAnthropology , 10(1):1–43, 1954.[40] Ruby and I. Kaur. A review of community detection algorithms in signed social net-works. In , pages 413–416, Aug 2017.3241] L. J. Rusnak, E. Robinson, M. Schmidt, and P. Shroff. Oriented hypergraphic matrix-tree type theorems and bidirected minors via boolean ideals.
J Algebr Com , pages 1–13,2018.[42] Stuart Russell and Peter Norvig.
Artificial Intelligence: A Modern Approach . PrenticeHall Press, USA, 3rd edition, 2009.[43] Andrew D. Selbst, Danah Boyd, Sorelle Friedler, Suresh Venkatasubramanian, andJanet Vertesi. Fairness and abstraction in sociotechnical systems. In
Proceedings ofACM Conference on Fairness, Accountability, and Transparency , 2018.[44] B. She, S. Mehta, C. Ton, and Z. Kan. Controllability ensured leader group selectionon signed multiagent networks.
IEEE Transactions on Cybernetics , 50(1):222–232, Jan2020.[45] B. She, S. Mehta, C. Ton, and Z. Kan. Controllability ensured leader group selection onsigned multiagent networks.
IEEE Transactions on Cybernetics , 50(1):222–232, 2020.[46] J. Tang, Y. Chang, C. Aggarwal, and H. Liu. A survey of signed network mining insocial media.
ACM Computing Surveys , 49(3), 2016.[47] Jelena Teˇsi´c, Joshua Mitchell, Eric Hull, and Lucas Rusnak. graphb: Python softwarepackage for graph analysis at scale. https://github.com/DataLab12/graphB , 2020.[48] W. T. Tutte.
Graph theory , volume 21 of
Encyclopedia of Mathematics and its Appli-cations . Addison-Wesley Publishing Company Advanced Book Program, Reading, MA,1984. With a foreword by C. St. J. A. Nash-Williams.[49] Stanley Wasserman and Katherine Faust.
Social network analysis: Methods and appli-cations , volume 8. Cambridge university press, 1994.[50] David Bruce Wilson. Generating random spanning trees more quickly than the covertime. In
Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Com-puting , STOC âĂŹ96, page 296âĂŞ303, 1996.[51] W. Yuan, K. He, D. Guan, and G. Han. Edge-dual graph preserving sign prediction forsigned social networks.
IEEE Access , 5:19383–19392, 2017.[52] Thomas Zaslavsky. Signed graphs.
Discrete Appl. Math. , 4(1):47–74, 1982. MR84e:05095a. Erratum, ibid., 5 (1983), 248. MR 84e:05095b.[53] X. Zhao, X. Liu, and H. Chen. Network modelling and variational bayesian inferencefor structure analysis of signed networks.
Applied Mathematical Modelling , 61:237–254,2018.[54] J. Zhou, L. Li, A. Zeng, Y. Fan, and Z. Di. Random walk on signed networks.