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Generalization of Effective Conductance Centrality for Egonetworks

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Generalization of Effective Conductance Centrality for Egonetworks Generalization of Effective Conductance Centrality for Egonetworks∗ Heman Shakeri1, Behnaz Moradi-Jamei2, Pietro Poggi-Corradini3, Nathan Albin3, and Caterina Scoglio1 1Electrical and Computer Engineering Department, Kansas State University, Manhattan, Kansas, USA 2Department of Statistics, Kansas State University, Manhattan, Kansas, USA and 3Department of Mathematics, Kansas State University, Manhattan, Kansas, USA (Dated: October 9, 2018) We study the popular centrality measure known as effective conductance or in some circles as information centrality. This is an important notion of centrality for undirected networks, with many applications, e.g., for random walks, electrical resistor networks, epidemic spreading, etc. In this paper, we first reinterpret this measure in terms of modulus (energy) of families of walks on the network. This modulus centrality measure coincides with the effective conductance measure on simple undirected networks, and extends it to much more general situations, e.g., directed networks as well. Secondly, we study a variation of this modulus approach in the egocentric network paradigm. Egonetworks are networks formed around a focal node (ego) with a specific order of neighborhoods. We propose efficient analytical and approximate methods for computing these measures on both undirected and directed networks. Finally, we describe a simple method inspired by the modulus point-of-view, called shell degree, which proved to be a useful tool for network science. The concept of information centrality was first intro- where σ(e) > 0, is the conductance of the edge e. Thus duced in [1] and was later reinterpreted in terms of elec- modulus is a constrained convex optimization problem trical conductance in [2]. Given a network G = (V; E) that has a unique extremal density ρ∗ when 1 < p < . and a node a V , the effective conductance centrality of This point of view allows for much more flexibility, be-1 a is defined as2 cause it can be applied to a variety of different families of X 1 objects: walks, cycles, trees, etc, and also works when the eff(a) := : (1) underlying network is directed or weighted. Moreover, C eff(a; b) b V a 2 n R modulus has very useful properties of Γ-monotonicity and countable subadditivity. where (a; b) is effective resistance distance between a eff For undirected networks the effective conductance be- and b.R Note that this measure considers every possible tween a and b is connected to Mod (Γ(a; b)) as follow path that electrical current flow might take from a to an 2 [6, 7] arbitrary sink b. The situation can be clarified by introducing the no- 1 = Mod (Γ(a; b)): (5) tion of modulus of families of walks. This is a way of (a; b) 2 measuring the richness of certain families of walks on a Reff network (and beyond, see [3{5]). Given two nodes a and In the following, we reproduce a proof for this connec- b we may consider the connecting family Γ(a; b) of all tion and how to calculate Mod2(Γ(a; b)) in symmetric walks γ from a to b. Then, given edge density ρ : E R networks using the pseudoinverse of the Laplacian. ! for p [1; ], we define `ρ (Γ) := minγ Γ `ρ (γ) where Let F be the set of all unit flows f : E R that satisfy 2 1 2 ! `ρ(γ) is the ρ-length of a walk γ: Kirchoffs node law and pass through a network G from X a to b. Namely for v V `ρ (γ) := ρ (e) : (2) 2 8 e γ 1 v = a 2 <> The p-modulus of Γ is defined as ( :f)(v) = 1 v = b r − :>0 otherwise Modp (Γ) := min Energyp (ρ) (3) `ρ(Γ) 1 ≥ corresponds to the injected currents at each node. The Namely, we minimize the energy of candidate edge- arXiv:1705.02703v2 [physics.data-an] 26 Jul 2018 energy of f is densities ρ subject to the ρ-length of every walk in Γ X being greater than or equal one, i.e., `ρ(Γ) 1. These Energy(f) := (e)f(e)2 ≥ R densities can be interpreted as costs of using the given e E edge. The energy we consider is 2 X where (e) = 1 is the resistance of edge e. A unit Energy (ρ) = σ(e)ρ (e)p ; (4) R w(e) p current flow i F is a unit flow that also satisfies Ohm's e E 2 2 law, i.e., there is a function V : V R (called a poten- tial) such that for every edge (a; b):! ∗ Corresponding author: [email protected] (a; b)i(a; b) = V(b) V(a): R − 2 Let U : V R be a vertex potential function. We I. EGOCENTRIC EFFECTIVE CONDUCTANCE ! can define a density ρU as the gradient of U, i.e., for the CENTRALITY edge e = v; w f g U U As mentioned above, eff(a) is sociocentric in the sense ρU(e) := u w C j − j that it considers all walks from a to an arbitrary node in Then, ρU is admissible for walks from a to b, whenever G. However, in practice, it can be prohibitive to scale U(a) = 0; U(b) = 1. sociocentric methods to very large networks. Moreover, Conversely, if ρ is an admissible density, then we can in real-world situations it is not feasible to have access define a potential U(x) as the infimum of `ρ(γ) over all to the entire network. Rather, one can at best know walks from a to x. With this definition, ρU = ρ, see [7]. local information up to a few neighborhood levels. For In particular, assuming each edge has a unit resistance, instance, when data is anonymized to protect privacy of network entities, identifying the sociocentric picture is X 2 X Energy(ρU) = ρU(e) = U(u) U(w) : impossible, e.g., sexual networks may be limited to the j − j e E e= u;w E number of contacts of individuals. 2 f g2 An alternative approach is to consider measures that Hence, if we substitute U with V + C, where V eff(a;b) are adapted to egonetworks (also known as neighborhood is the electric potential when a unitR current flow i F networks). An ego network Ga(r) around a node a is 2 is passing through the network with source a and sink b constructed by collecting data (nodes and edges) start- and the effective resistance between a and b is eff, then, ing from the ego a and searching G out to a predefined R order of neighborhood r 1; (a) ; where (a) is the T 1 2 f ··· g Mod (a; b) = min ρUρU = : eccentricity of node a or the maximum distance from a 2 U (6) a=0 (a; b) U eff b=1 R to nodes in G. Egonetworks are often preferred because they support By Kirchhoff's law of current conservation: more flexible data collection methods [9] and often in- X volve less expensive computation costs. Egocentric mea- a (V V ) = ( :i)(i) i;j i − j r sures are more stable [10] against network sampling and j reliable (less sensitivity) with measurement errors [11]. N N where A = [aij] R × is the adjacency matrix of G, We concentrate on unweighted (binary) networks to sim- 2 with aij = 1 if and only if i; j E. In matrix form: plify the algebra, although, all of our methods and dis- 2 cussions can be easily generalized for weighted networks. LV = I (7) Thus, we let d(a; b) denote the shortest-path distance be- tween two nodes (smallest number of hops). The neigh- where L is the Laplacian matrix of G and I = :i. Be- borhood structure around an ego a is described by the cause V is defined up to an additive and the nullspacer of shells of order k: L is along the constant vector, we ground an arbitrary node k and thus reduce L by removing kth row and col- S(a; k) := y V : d(a; y) = k ; umn denoted by kL [8]. Now we can find solve (7): f 2 g and the corresponding families of walks Γ(v; S(a; k)), k k 1 k V = ( L)− I: consisting of simple walks that begin at ego v V and reach S(a; k) for the first time. Modulus allows2 a quan- k 1 we denote ( L)− by (reduced conductance matrix) tification of the richness of the family of walks, i.e., a G and obtain effective resistance between nodes a and b is family with many short walks has a larger modulus than k k a family with fewer and longer walks. Here we consider eff(a; b) = Va Vb R − (8) shell modulus Mod2(v; S(a; k)) which quantifies the ca- = + 2 Ga;a Gb;b − Ga;b pacity of walks emanating from the ego up to the shell and from (6): S(a; k) [5] without having to account the data outside Ga(k). 1 Mod (a; b) = ( + 2 )− (9) 2 Ga;a Gb;b − Ga;b Theorem 1. For undirected networks, we can calcu- Therefore, using (5), we can rewrite the effective con- late 2-modulus of Γ(v; S(a; k)) analytically without going ductance centrality in (1) in the Modulus language through the optimization problem in (3): PSs 1 1 X 1 + xs − (a) = Mod (Γ(a; b)): (10) j=S1 xi Ceff 2 Mod2(a; S(a; r)) = (11) b V a xs 2 n PSs 1 For the rest of this paper, we consider p = 2 due to its where xi = − ij .
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