Improving In-Network Computing in IoT Through Degeneracy
Merim Dzaferagic, Neal McBride, Ryan Thomas, Irene Macaluso, Nicola Marchetti
11 Improving In-Network Computing in IoT Through Degeneracy
Merim Dzaferagic, Neal McBride, Ryan Thomas, Irene Macaluso, and Nicola Marchetti
Abstract —We present a novel way of considering in-networkcomputing (INC), using ideas from statistical physics. We definedegeneracy for INC as the multiplicity of possible optionsavailable within the network to perform the same function witha given macroscopic property (e.g. delay). We present an efficientalgorithm to determine all these alternatives. Our results showthat by exploiting the set of possible degenerate alternatives, wecan significantly improve the successful computation rate of asymmetric function, while still being able to satisfy requirementssuch as delay or energy consumption.
Index Terms —In-network computing, distributed computing,degeneracy, redundancy, IoT
I. I
NTRODUCTION
Internet of Things (IoT) networks perform actions or makedecisions powered by access to raw or processed data gath-ered by spatially distributed “things”. Distributed in-networkcomputation (INC) approaches, i.e. the processing of raw datawithin the IoT network, are becoming increasingly popularin that they can potentially achieve higher energy efficiency,lower computational delay, and higher robustness compared tothe traditional approach, which involves transmitting raw datato the sink and then performing the computation.As IoT networks become increasingly large, their inter-communication becomes complex enough as to resemblea thermodynamic system of many interacting objects. Thissituation leads naturally to a physics-inspired study of theirbehavior using statistical mechanics and a concept knownas degeneracy. Degeneracy arises from structurally differentconfigurations of a system (microstates) having functionallysimilar/identical macroscopic properties (macrostates). A con-cise definition of degeneracy comes from a modern reviewof the use of the term: “degeneracy describes the ability ofdifferent structures to be conditionally interchangeable in theircontribution to system functions” [1]. In the context of INC,we view these different structures as functional topologies (FT)[2]: subsets of the network which enable the computation ofa distributed function. In this paper, we consider degeneracyas the multiplicity of available FTs that enable the distributedINC with a given macroscopic observable property i.e. delay.The intended application of an IoT network usually definesthe type of functions being considered. This in turn naturallydefines the observables (macrostates) of interest to analyze
This publication has emanated from research supported in part by a researchgrant from Science Foundation Ireland (SFI) and is co-funded under theEuropean Regional Development Fund under Grant Number 13/RC/2077 andin part by work supported by the Air Force Office of Scientific Research underaward number FA9550-17-1-0066.Merim Dzaferagic, Neal McBride, Irene Macaluso and Nicola Marchetti arewith CONNECT, Trinity College Dublin, Ireland. (email: [email protected],[email protected], [email protected], [email protected])Ryan Thomas is with AFOSR, Air Force Office of Scientific Research,USA. (email: [email protected]) these functions. The distributed computing literature [3]–[7]usually assumes an error free scenario, i.e. no node failures,and mainly focuses on the optimization of the computationalrate and the communication cost. On the other hand, we areinterested in a more realistic network setup, in which nodefailures do exist, and we study the impact of degeneratecomputation and communication paths on the success rate ofthe computation.Authors like [3]–[9] address communication aspects ofdistributed in-network computation. They focus on the opti-mization of the communication/computation parameters, likecomputational complexity, energy efficiency, computationalthroughput and computational delay. In contrast, we focus onthe degeneracy of the network by finding the multiple feasiblecomputational graphs that satisfy a given requirement (e.g.delay), rather than a single optimal computational graph. Thisway, it is possible to fully harness the computational capabilityof the network and significantly increase the robustness ofthe computation, while still being able to satisfy require-ments, such as delay or energy consumption. Even thoughthe INC literature features analysis of functions ranging fromdistributed neural networks [8] to very complex computationalframeworks like MapReduce and Dryad [5], the majority ofIoT applications, however, rely on simple aggregate functionslike max, min, count, sum [3]. The distributed computationof these functions is usually modeled with tree structures, e.g.Steiner trees [3], [4]. Closely related to the approach proposedin our work is the standard Steiner Tree Packing (STP)problem [10] and its well know relaxation, i.e. the FractionalSteiner Tree Problem. Instead of using the standard STP, werather extend the search from minimum weight Steiner treesto all existing Steiner trees in which the sum of the weightson the links is lower than a predefined value.The main contributions of this work are: an efficient al-gorithm that generates all functional topologies satisfying agiven delay requirement, for any physical topology; a for-mal definition and calculation of degeneracy and the relatedconcept of redundancy in terms of distributed computingover IoT networks; a comparison of degenerate INC and atraditional multi-hop scheme that selects a single graph forthe computation showing that it is possible to significantly in-crease the probability of successful computation by exploitingdegeneracy. II. F RAMEWORK
We define a physical network G = ( V , E ) , as a simple graphthat contains no self-edges or repeated edges, where the set The Steiner Tree Packing problem is to find the maximum number ofedge-disjoint subgraphs of a given graph that connect a given set of requiredpoints. a r X i v : . [ c s . D C ] M a y of nodes V represents the physical nodes in the network (e.g.sensors in an IoT network), and the set of edges E representsthe physical links between the nodes that can interact directlywith each other. Mathematical functions, f , can be representedas subgraphs H of the overall network which we refer toas FTs. Even though the definition of FTs is much broader[2], in terms of INC the FTs are directed, rooted trees, i.e.Steiner trees, with all edges pointing towards the root, Y . Thisroot/sink node is where the result of a particular function f must reach. The leaves of this in-tree are the inputs of thefunction, X . These inputs may take the form of entries in adistributed database or measurements of the environment suchas temperature. The set of inputs, x i ∈ X , are generated innodes v i .The individual operations of a given function, f ( X ) , aremapped to specific nodes in the network. Each FT, H , is somesubset of U ⊂ V and D ⊂ E in which each node and edge isinvolved in computing and routing f ( X ) . Multiple, unique FTswhich model the same function, f ( X ) , can be chosen from thesame physical topology, G . Any distinct FTs that perform thesame calculation and result in the same observable macrostateof the network function are considered to be degenerate. Theoverall delay associated with a given FT is the principalmacrostate of interest in this paper. We define the delay asthe maximum graph distance between the sink node, Y , andany other node, u ∈ H . Alternative macrostates of potentialinterest are energy efficiency, which could be approximated bythe number of edges in FT H , and computational throughput(number of function calculations per unit delay).In [11], the authors made the distinction between two classesof network functions, namely divisible and indivisible. Denotea set W = { , . . . , | W |} and a subset S = { i , . . . , i k } ⊂ W ,where i , < i , < . . . , < i k . Let x S be a set { x i , x i , . . . , x i k } .A function, f : x S → y , is divisible if given any partition of S , Π ( S ) = { s , . . . , s j } of S ⊂ W , there exists a function g Π ( S ) such that for any x S , f ( x S ) = g Π ( S ) ( f ( x s ) , . . . , f ( x s k )) . (1)Otherwise, f is indivisible. A. Degeneracy of Functional Topologies
Degeneracy can be considered in terms of these two classesof network functions. We define strong degeneracy to bethe multiplicity of FTs of an indivisible network function.We define weak degeneracy to be the multiplicity of FTsof divisible functions. The aggregate functions mentioned inSec I are all examples of divisible functions. Due to thepreviously discussed importance of these aggregate functionsfor distributed computing, we focus on the aspects of weakdegeneracy in this paper.For a physical topology, G , a network function, f , and aninput & output set of nodes, X & Y , we can define weakdegeneracy of the physical topology, G , with respect to anobservable (e.g., delay, energy efficiency) as the multiplicityof FTs, g W ( G , X , Y , f , d ) = |{ H ( G , X , Y , f , d )}| , (2) with a given delay, d . For completeness, strong degeneracy is defined as the multiplicity of FTs on top of a physicaltopology G with respect to an observable, with input nodes X ,performing an indivisible function in which the sub-operationsmust be performed in an exact ordering, g S ( G , X , Y , f , d ) .Since f ( X ) is divisible, we can perform the calculationusing any partition s of X and combine the result of eachsubset f ( x s ) in any order. Weak degeneracy of a full meshphysical topology is therefore dependent on the number ofways of partitioning a set, known as the Bell number, B n + = n (cid:213) k = (cid:18) nk (cid:19) B n , (3)for a set of n + elements and B = B = . It shouldbe noted that the Bell number is an upper bound for theweak degeneracy. The weak degeneracy of a generic physicaltopology has to further account for the number of mappings ofthese partitions to a given FT. This is the number of ways wecan route each partition such that the suboperation f ( x s i ) onpartition s i occurs when the elements of partition s i intersectfor the first time.Our degeneracy analysis allows us to both identify the setof subgraphs which perform a function and to sort these withregard to observables like delay, energy efficiency or com-putational throughput. This insight may be used in future toimprove computational throughput, computational resilience,deploy superior network configurations, or to analyze thedegeneracy potential of a routing protocol, i.e., how many ofthe feasible FTs can a routing protocol discover. B. Redundancy of Functional Topologies
Degeneracy arises from structurally different FTs havingfunctionally identical properties. We can also define redun-dancy in FTs, which is related to degeneracy, with tworedundant FTs not sharing any common nodes or edges apartfrom their common X and Y . The redundancy between twoFTs, r ( H a , H b ) , is defined as r ( H a , H b ) = (cid:40) , if U a ∩ U b = X ∪ Y , , otherwise. (4)We define the average redundancy, R ( G , Y , f ) = |{ X }| (cid:213) { X } N X (cid:213) { H i , H j } i (cid:44) j r ( H i , H j ) , (5)the mean over the number of FTs which are redundant to anyother given FT. The sum is over all N X possible FTs and theset of all possible input sets, { X } . The total redundancy gives ameasure of the number of ways to choose alternate FTs whichbypass nodes in case of node/link failure. C. Find all weakly degenerate functional topologies
In the context of INC, an FT describes the interactions be-tween network nodes in the course of computing a distributedfunction [2]. For the INC of divisible functions FTs are Steinertrees, i.e. they include only the nodes and edges from the
Physical topologySet of paths between the sink and target nodesFunctional/Computationalgraph
Fig. 1. Process to identify all FTs from a generic physical topology. physical topology that are involved in the computation and/orrouting of the function.To generate an FT, consider a partition S of the input set X . For each subset s ∈ S , the function f acts on the elementsin s at their first meeting point node in H . For a subset s = { x , x } , we construct paths, p = x , v , . . . , v i , . . . , Y and p = x , u , . . . u k , . . . , Y . The sub-operation f ( x , x ) occurs wherethese paths meet for the first time, v l = u k . Paths p and p must contain at least one crossing point since they havea common root at Y . Having met at u k = v l , the result ofsub-operation f ( x , x ) can be routed down either sub-path to Y . If there exist further subsets s in the partition, these aretreated in the same manner. The order in which we combinepaths between any subsets, s , and Y does not matter since f is divisible. procedure FIND FT S ( G , X , Y , f , d max ) Find all directed, simple paths, { p x i } , from all sourcenodes, X , to the target node, Y of length d max or less. (cid:46) e.g. Breadth-First Search Take the union of edges and nodes of all possiblecombinations, C , of paths from different input nodes p x i ∀ x i . if A given combination, c ∈ C , is not a tree (containsone or more cycles). then Find all unique spanning trees of c . for All unique spanning trees do if Spanning trees has leaf nodes, L (cid:42) X . then Remove leaf nodes, L , since they are notan input. end if end for end if Remove any duplicate FTs. end procedure
Fig. 2. Our algorithm to generate all FTs of a graph G , for a symmetricfunction f , with input nodes X , sink node, Y , and a maximum delay cutoff d max . If we consider a specific path each from x and x to Y and take the union of the nodes and edges in each path, wegenerate a subgraph of G rooted at Y and with leaves X . Ifthe union of the paths join and later diverge, one or moreundirected cycles have been formed. Each independent patharound the cycle results in a different FT. These FTs can be identified by finding the unique spanning trees of the cycle-containing subgraphs and removing the resulting leaves whichare not inputs, X , since they don’t perform a computational orrouting role.We assume that G is connected and there exist paths fromeach input node to the sink. To aid in simulation, we restrictthe maximum path length (delay), d max , to the maximum graphdistance between the sink and any other node, known asthe eccentricity. Figure 1 depicts this process of extractingthe FTs from a physical topology and Fig. 2 outlines ouralgorithm. Our algorithm extends the standard Steiner TreePacking problem and its relaxation, i.e. the Fractional SteinerTree Problem (which are known to be NP-hard [10]), byextending the search from minimum weight Steiner trees toall existing Steiner trees in which the sum of the weightson the links is lower than a predefined value. To the bestof our knowledge, this problem has not been addressed in theliterature so far. III. A NALYSIS
We now present the numerical analysis of the degeneracyand redundancy of two different physical topologies: an × square lattice and a randomly placed sensor network withthe same number of nodes. In the lattice case, we place thesink node, Y , in the centre of the lattice and we restrict themaximum delay to hops, equal to the eccentricity of thesink node. In the case of a random topology, we consideran area of × km, place the sink node at the centre ofthe area, and randomly distribute the remaining nodes.The edges are chosen to connect each node to its four closestneighbors . The simulation involves calculating all FTs forrandomly sampled input node pairs, X , which are chosento be within three hops from each other, since in-networkcomputation is typically used for data collected by nearbysensors. To calculate the average number of degenerate FTswith a given delay, this process is repeated times. Thestandard errors on the means are estimated using bootstrapresampling of all simulations, and are in-fact smaller than theplot markers.Shown in Fig. 3 are the cumulative degeneracy of FTs andcumulative average redundancy for both topologies. For thelattice case, the cumulative number of degenerate FTs (bluemarkers) is seen to increase exponentially with the increasingdelay limit. This same exponential behaviour has also beenseen in independent experiments on different lattice sizes.It is a result of the branching process associated with FTsof increasing size gaining access to more and more nodesand their respective edges. The cumulative redundancy ofFTs (red markers) also increases exponentially with delaylimit. Although the cumulative degeneracy and redundancyare lower in the case of a random topology than thoseobserved in a lattice, they still increase exponentially as thedelay limit increases. The increase in cumulative redundancydemonstrates that the ability for physical topologies to performparallel computing using independent FTs is higher for lessrestrictive delay limits. It is worth noting that this results in a non-uniform distribution of theedges.
Fig. 3. Log plot (base 10) of cumulative degeneracy and cumulative redun-dancy. For the square lattice, the slopes of the linear fits for the cumulativedegeneracy and cumulative redundancy are: 0.69 and 0.59, respectively. Forthe random topology, the slopes of the linear fits for the cumulative degeneracyand cumulative redundancy are 0.51 and 0.36, respectively.
Using the numerical estimates of degeneracy, we now ex-amine the computational robustness of INC. Let us first definethe probability of computational success, α i , at node i as thenumber of successful computations as a fraction of attempts.The probability of a node failing during the computation of afunction is then found to be, P f i = P c i · ( − α i ) , (6)where, P c i , is the probability of a node occurring in anyfeasible FT. A computation may fail at a node for a numberof reasons such as excessive load or during a sleep cycle. Theprobability of a successful computation is given as P S c = (cid:214) i ∈ U ( − P f i ) , (7)where U are the nodes in the FT that performs the com-putation. Figure 4 compares the probability of successfulcomputation using a single FT, e.g. selected as the optimalcomputational graph in terms of delay or energy consumption,and a random selection of one of the multiple FTs that canperform the same computation while satisfying a maximumdelay requirement ( for the results in figure). In the caseof a single optimal FT, P c i = for all nodes, since allnodes in the optimal graph must be used for INC. Consideringthe degeneracy of FTs from Fig. 3, each computation canpotentially be performed by using any of the degenerate FTs.Therefore, we estimate P c i for the randomly selected FT,that is going to perform the computation, as the frequencyof occurrences of the node i in all the degenerate FTs. This P c i is then used to compute the corresponding P S c accordingto (6) and (7).Results are then averaged over all the input pairs. Theresults in Fig. 4 show that it is possible to significantly increasethe probability of successful computation by exploiting themultiplicity of FTs, i.e. the degeneracy.IV. C ONCLUSION
We presented a novel way of considering INC using ideasfrom statistical physics. In particular, we defined degeneracyand the related concept of redundancy for distributed computa-tion in networks, and we introduced an algorithm to efficiently
Fig. 4. Probability of successful computation vs number of nodes involvedin the computation. The blue dots refer to a single FT; the green and orangedots refer to a randomly selected FT that can perform the same function forthe square lattice and random topology respectively. The delay limit for allcalculations is set to be equal to . compute all degenerate and redundant functional topologies.Our results show that the cumulative degeneracy and re-dundancy of functional topologies increase exponentially asthe accepted delay limit for the computation is increased.The successful computation rate of a symmetric function isshown to be significantly higher using the set of possibledegenerate functional topologies as opposed to exclusivelyusing the optimal functional topology. This means that we canconsiderably improve the robustness of the computation, whilestill being able to satisfy requirements, such as delay or energyconsumption. Future work will focus on exploiting degenerateand redundant FTs to perform parallel computing. The caseof redundant, i.e. independent FTs, is of particular interest inthat bottlenecks can be avoided with no coordination betweenthe nodes of different FTs.R EFERENCES[1] P. H. Mason, “Degeneracy: Demystifying and destigmatizing a coreconcept in systems biology,”
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