Temporal graphs.

Definitions, problems, algorithms, and complexity

• Temporal graphs
  • Graphs that change with time
  • Give rise to new set of challenging, important, and practical problems
• Formal treatment in its infancy
• Big set of applications
  • Information and communication networks
    • Mobile ad hoc, sensor, peer-to-peer networks
  • Social networks
    • Social relationships
  • Transportation networks
  • Physical systems
    • Molecule interactions
  • Epidemics
    • Contacts

• Removing all time information from the temporal graph
  • Collapsing (if necessary) multiple edges between any two vertices into a single edge
• Loss of all the temporal information
  • Critical to understanding relationships between entities
    
    
    
    
    
    
     
    • Paths from 1 to 2 in non-temporal do not exist in temporal
    • Path of length 2 from 1 to 5 in non-temporal does not exist in temporal

• Complexity of optimisation problems
  • Dynamic languages derived from NP-complete languages become PSPACE-complete
    • Not equivalent model
• Classical results
  • Max-flow min-cut
  • Menger's theorem
• From instantaneous properties to global properties
  • Connectivity, diameter

• Several different definitions of temporal graphs with different names
  • Evolving graphs
    • Sequence of static graphs
  • Time-varying graphs
    • Static graph with labelling on the edges assigning set of natural numbers (availability period)
  • Link stream graphs
    • Static multi-graph with labelling on the edges assigning natural number (availability time)
  • Dynamic graphs
    • Probabilistic process specifying the next graph

• Discrete temporal graph \(\lambda(G)\)
  • Graph \(G=(V,E)\)
  • Labelling \(\lambda:E\rightarrow 2^\mathbb{N}\)
• Notations
  • \(\lambda(E)\): multi-set of all labels
  • \(|\lambda|=\sum_{e\in E}|\lambda(e)|\)
  • \(\lambda_\mathrm{max}=\max\{l\in\lambda(E)\}\) and \(\lambda_\mathrm{min}=\min\{l\in\lambda(E)\}\)
  • Lifetime: \(\alpha(\lambda)=\lambda_\mathrm{max}-\lambda_\mathrm{min}+1\)
• ​Snapshot \(\lambda(G)[t]\) (simply \(G_t\))
  • ​\(G_t=(V,E_t)\) where \(E_t=\{e\in E:t\in\lambda(e)\}\)
• ​Temporal graph viewed as sequence of static graphs
  • ​\(G_1,G_2,\ldots,G_{\lambda_\mathrm{max}}\)

• Static expansion of discrete temporal graph \(\lambda(G=(V,E))\)
  • Static graph \(H=(S,F)\)
    • \(S=\{v_{t,i} : v_i\in V \wedge \lambda_\mathrm{min}-1\leq i\leq\lambda_\mathrm{max}\}\)
    • \(F=\{(u_{{t-1},i},u_{t,j}) : \lambda_\mathrm{min}\leq i\leq\lambda_\mathrm{max}\wedge(i=j\vee(v_i,v_j)\in E_t)\}\)

• Temporal walk \(\mathbb{W}\) from \(u\) to \(v\)
  • Alternating sequence of nodes and times \((u_1,t_1,u_2,t_2,\ldots,u_{k-1},t_{k-1},u_k)\) s.t.
    • \(u_1=u\) and \(u_k=v\)
    • For each \(i\) with \(1\leq i<k\), \((u_i,u_{i+1})\in E_{t_{i}}\)
    • For each \(i\) with \(1\leq i<k-1\), \(t_i<t_{i+1}\)
  • Departure time: \(t_1\)
  • Arrival time: \(t_{k-1}\)
  • Duration: \(t_{k-1}-t_1+1\)
• Journey \(\mathbb{J}\) from \(u\) to \(v\)
  • Temporal walk from \(u\) to \(v\) with distinct nodes
• Foremost journey from \(u\) to \(v\) from time \(t\)
  • Departure time \(t\) and minimum arrival time
• Temporal distance from \(u\) to \(v\) at time \(t\)
  • Duration of foremost journey from \(u\) to \(v\) from time \(t\)

• Example

• Hypothesis

  • Entire temporal graph available

  • Edges sorted by their apparition time

• Algorithm similar to BFS

  1. Initialize \(\mathbf{t}[s]=t-1\) and \(\mathbf{t}[v]=\infty\) for all \(v\neq s\)

  2. For each edge \(e=(u,v)\)

     1. If \(t\leq\lambda(e) \wedge \mathbf{t}[u]<\lambda(e)<\mathbf{t}[v]\), then \(\mathbf{t}[v]=\lambda(e)\)

  3. Return \(\mathbf{t}-(t-1)\)

• Complexity \(O(\lambda(E))\)

• Example

• Menger's theorem

  • The maximum number of node-disjoint \(s\)-\(z\) paths is equal to the minimum number of nodes that must be removed in order to separate \(s\) from \(z\)

• Temporal (single-labeled) counter-example: \(s=v_1\) and \(z=v_5\)

• Every journey includes two inner nodes \(v_2\),\(v_3\), or \(v_4\)
  • If \(v_2,v_3\) or \(v_2,v_4\), vertical obstacle is created
  • If \(v_3,v_4\), only other path is \((v_1,v_2,v_5)\) which is not a journey
• Hence, no node-disjoint journey
• On the other hand, any two inner nodes are in a journey
• Hence, minimum separator has size 2
• Can be genralized to \(2k-1\) inner nodes

• Menger's theorem in temporal graphs

  • The maximum number of edge-disjoint \(s\)-\(z\) paths is equal to the minimum number of edges that must be removed in order to separate \(s\) from \(z\)

• Proof (reduction to Menger's theorem on static directed graphs)

  • Step 1

• Menger's theorem in temporal graphs

  • The maximum number of edge-disjoint \(s\)-\(z\) paths is equal to the minimum number of edges that must be removed in order to separate \(s\) from \(z\)

• Proof (reduction to Menger's theorem on static directed graphs)

  • Step 2

• Preserves edge-disjointness and sizes of separators

• Fact

  • Any temporal clique is temporally connected

    • For any pair of nodes \(u\) and \(v\) there exists a journey from \(u\) to \(v\)

• Temporal spanner

  • Subset of temporal edges maintaining connectivity

• Given a connected temporal graph, find a "sparse" temporal spanner (if it exists)

• Temporal hypercube

  • Labels are positions in which two binary strings differ

• No "sparser" temporal spanner

• Temporal cliques with just one label

  • No spanner of size \(o(n^2)\)

• Question
  • What about temporal cliques with all distinct labels?

• Dismountable temporal cliques

  • There exists node \(v\) with two neighbours \(u\) and \(w\) such that

    • \(\lambda(v,u)=\min_x\lambda(x,u)\)

    • \(\lambda(v,w)=\max_x\lambda(x,w)\)

• Dismountable temporal cliques

  • There exists node \(v\) with two neighbours \(u\) and \(w\) such that \(\lambda(v,u)=\min_x\lambda(x,u)\) and \(\lambda(v,w)=\max_x\lambda(x,w)\)

  • ​If \(S\) is a temporal spanner for \(G\setminus v\), \(S\cup\{(v,u),(v,w)\) is a temporal spanner for \(G\)

• Algorithm

  • While \(G\) is dismountable for nodes \(v,u,w\)

    • \(S\): output of algorithm on \(G\setminus v\)

    • \(S = S\cup\{(v,u),(v,w)\}\)

• If algorithm reduces to one temporal edge, returned temporal spanner has linear size

  • At each step, it adds two edges to the temporal spanner

• Unfortunately there are infinite temporal cliques which are not dismountable

\(4\) nodes

\(8\) nodes

• Any temporal clique with distinct labels admits a temporal spanner of size \(O(n\log n)\)

  • Clever usage of delegation of emissions and receptions to neighbours

  • Recursive sparsification technique

• Open problem

  • Does a linear size temporal spanner always exist?

• Given temporal \(G=(V,E)\)
  • \(E\): set of temporal edges \((u,v,t)\) for any \(t\in\lambda(u,v)\)
• \(S\subseteq E\)  is \(s\)-\(d\) disconnecting set of \(G\) if there is no \(s\)-\(d\) journey in \(G\setminus S\)
  • \(G\setminus S = (V,E\setminus S)\)
• \(S\) is an \(s\)-\(d\) cutset if it is a minimal \(s\)-\(d\) disconnecting set
  • No proper subset of \(S\) is an \(s\)-\(d\) disconnecting set 

\(s\)-\(d\) disconnecting set

\(s\)-\(d\) cutset

• Given temporal \(G=(V,E)\), simplified static expansion \(G^T=(V^T,E^T)\)
  • \(V^T=\bigcup_{e\in E}\{u^e, {}^{e}v : e=(u,v,t)\}\)
  • \(E^T = \bigcup_{e\in E}\{(u^e, {}^{e}v)\} \cup \bigcup_{e,f\in E}\{({}^{e}u, u^{f}) : e=(x,u,t_e)\wedge f=(u,y,t_f)\wedge t_f>t_e\}\)

• Given temporal \(G=(V,E)\) and \(X\subseteq V^T\), cut \(\langle X,\bar{X}\rangle\) in \(G\)
  • \(\langle X, \bar{X}\rangle = \{e\in E : e=(u,v,t)\wedge u^e\in X \wedge {}^{e}v\in\bar X\}\)

• Given \(G=(V,E)\), \(s,d\in V\) and \(X\subseteq V^T\), \(X\) is \(s\)-\(d\) good if
  • \(\forall_{e=(s,x,t)\in E}[s^e\in X]\) and \(\forall_{e=(x,d,t)\in E}[{}^{e}d\not\in X]\)
  • \(u^{e=(u,x,t_e)}\in X \Rightarrow\forall_{f\in E : f=(u,x,t_f)\wedge t_f>t_e}[u^f\in X]\)
• ​If \(X\) is good, then \(\langle X,\bar{X}\rangle\) is \(s\)-\(d\) disconnecting set

• Given \(G=(V,E)\), \(s,d\in V\) and \(S\subseteq E\), \(S\) is \(s\)-\(d\) cutset iff
  • \(S=\langle X, \bar X\rangle\) for some \(s\)-\(d\) good set \(X\subseteq V^T\)
  • \(x\in X\) iff \(x\) is reachable in \(G^T[X]\) from \(s^{e}\) for some \(e\in E\)
  • if \({}^{e}u\in\bar{X}\) for some \(e=(x,u,t)\in E\) such that \(x^{e}\in X\), then, for some \(f\in E\), \({}^{f}d\) is reachable in \(G^T[\bar{X}]\) from \({}^{e}u\)​

• Footprint of \(\lambda(G=(V,E))\): \(G=(V,E)\)
• Eventual footprint of \(\lambda(G=(V,E))\): \(G=(V,E\setminus M)\) where
  • ​\(M=\{e\in E: \exists t^*\forall t>t^*[t\not\in\lambda(e)]\}\)

• \(\mathcal{TC}^\mathcal{R}\)
  • Set of temporal graphs such that any pair of nodes is able to communicate infinitely often through journeys
• Claim
  • \(\mathcal{TC}^\mathcal{R}\) is equivalent to the set of temporal graphs whose eventual footprint is connected
• ​Solving combinatorial optimisation problems on \(\mathcal{TC}^\mathcal{R}\)
  • ​Nodes do not know which edges are recurrent
  • Best they can do, compute solutions for the footprint and hope it will be a solution also in the eventual footprint

• A property \(P\) is robust in a (static) graph \(G\) iff it is satisfied in any connected spanning subgraph of \(G\)
  • Special case of (decreasing monotone) hereditary property
  • Never studied before
    • Quite natural
• Example: maximal independent set (MIS)
  • Robust MIS (RMIS) in \(G\): MIS in any connected spanning subgraph of \(G\)

Robust MIS

Non robust MIS

• All MIS are robust in \(G\) iff
  • \(G\) is complete bipartite
  • \(G\) is a sputnik
    • ​Every node in a cycle has a pendant neighbour
• Polynomial-time algorithm for deciding whether a graph \(G\) admits a robust MIS